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Enumeration problems in directed walk models Wong, Thomas 2015

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Enumeration Problems in Directed Walk Models.byThomas WongB. Science, The University of Queensland, Australia, 2007M. Science, The University of British Columbia, Canada, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University Of British Columbia(Vancouver)August 2015c© Thomas Wong, 2015AbstractSelf-avoiding walks appear ubiquitously in the study of linear polymers as it nat-urally captures their volume exclusion property. However, self-avoiding walks arevery difficult to analyse with few rigourous results available.In 2008, Alvarez et al. [1] determined numerical results for the forces inducedby a self-avoiding walk in an interactive slit. These results resembled the exactresults for a directed model in the same setting by Brak et al. [10], suggesting thephysical consistency of directed walks as polymer models. In the directed walkmodel, three phases were identified in the infinite slit limit as well as the regionsof attractive and repulsive forces induced by the polymer on the walls.Via the kernel method [8], we extend the model to include two directed walks asa way to find exact enumerative results for studying the behaviour of ring polymersnear an interactive wall, or walls.We first consider a ring polymer near an interactive surface via two friendlywalks that begin and end together along a single wall. We find an exact solution andprovide a full analysis of the phase diagram, which admits three phase transitions.The model is extended to include a second wall so that two friendly walksare confined in an interactive slit. We find and analyse the exact solution of twofriendly walks tethered to different walls where single interactions are permitted.That is, each walk interacts with the wall it is tethered to. This model exhibitsrepulsive force only in the parameter space. While these results differ from thesingle polymer models, they are consistent with Alvarez et al. [1].Finally, we consider the model with double interactions, where each walk in-teracts with both walls. We are unable to find exact solutions via the kernel method.Instead, we use transfer matrices to obtain numerical results to identify regions ofiiattractive and repulsive forces. The results we obtain are qualitatively similar tothose presented in Alvarez et al. [1]. Furthermore, we provide evidence that thezero force curve does not satisfy any simple polynomial equation.iiiPrefaceChapter 2 and Chapter 3 are prepared by Thomas Wong and contains the terminol-ogy, definitions, and introductory examples for methods used in the thesis.The results of Chapter 4 were originally due to Brak et al. [10]. It is reformu-lated using the methods presented in the thesis by Thomas Wong.Chapter 5 is adapted from a manuscript [55] published in Journal of Physics A:Mathematical and Theoretical.A. L. Owczarek, A. Rechnitzer, and T. Wong. Exact solution of two friendly walksabove a sticky wall with single and double interactions. Journal of Physics A:Mathematical and Theoretical, 45(42):425003, 2012Section 5.1 through Section 5.4 are work primarily done by Thomas Wong,while Section 5.5 and Section 5.6 are primarily due to Aleks Owczarek and AndrewRechnitzer with input from Thomas WongChapter 6 is adapted from a manuscript [74] published in Journal of Physics A:Mathematical and Theoretical.T. Wong, A. L. Owczarek, and A. Rechnitzer. Confining multiple polymers be-tween sticky walls: a directed walk model of two polymers. Journal of Physics A:Mathematical and Theoretical, 47(41):415002, 2014Chapter 7 is work by Thomas Wong with input from Andrew Rechnitzer. Theidentification of the research projects was made by Andrew Rechnitzer. Writing ofthe thesis is by Thomas Wong, with feedback and suggestions provided by AndrewRechnitzer.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Transfer Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Connection to Statistical Mechanics . . . . . . . . . . . . . . . . 253 Directed Walk Above a Wall . . . . . . . . . . . . . . . . . . . . . . 293.1 Non-Interacting Model . . . . . . . . . . . . . . . . . . . . . . . 293.2 Interacting Model . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Alternate Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Properties of Free Energy . . . . . . . . . . . . . . . . . . . . . . 443.4.1 Single Interaction Parameter . . . . . . . . . . . . . . . . 444 Directed Walk in the Slit . . . . . . . . . . . . . . . . . . . . . . . . 55v4.1 Non-Interacting Model . . . . . . . . . . . . . . . . . . . . . . . 564.2 Interacting Model . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2.1 Case (I): a = 1,b = 1 . . . . . . . . . . . . . . . . . . . 664.2.2 Case (II): a = 2,b = 2 . . . . . . . . . . . . . . . . . . . 674.2.3 Case (III): a = 2,b = 1 . . . . . . . . . . . . . . . . . . . 674.2.4 Case (IV): a,b < 2 . . . . . . . . . . . . . . . . . . . . . 684.2.5 Case (V): a < 2,b = 2 . . . . . . . . . . . . . . . . . . . 694.2.6 Case (VI): a > 2 and a > b . . . . . . . . . . . . . . . . . 704.2.7 Case (VII): a > 2 and a = b . . . . . . . . . . . . . . . . 704.2.8 Case (VIII): ab−a−b = 0 . . . . . . . . . . . . . . . . . 714.2.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 724.3 Properties of Free Energy (Continued) . . . . . . . . . . . . . . . 744.3.1 Multiple Interaction Parameters . . . . . . . . . . . . . . 745 Two Directed Walks Above a Wall . . . . . . . . . . . . . . . . . . . 785.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . 825.3 Solution of the Functional Equations . . . . . . . . . . . . . . . . 865.3.1 Solution of the Functional Equations When a = 1 . . . . . 865.3.2 Solution of the Functional Equation When a 6= 1 . . . . . 905.4 Alternate Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4.1 Solutions at a = 0,1 and 2 . . . . . . . . . . . . . . . . . 985.5 Analysis of Phase Structure and Transitions . . . . . . . . . . . . 995.5.1 Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.5.2 Desorbed to a-rich Transition . . . . . . . . . . . . . . . 1005.5.3 Desorbed to d-rich Transition . . . . . . . . . . . . . . . 1005.5.4 a-rich to d-rich Transition . . . . . . . . . . . . . . . . . 1025.5.5 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . 1035.5.6 Asymptotics of the d-rich- a-rich Phase Boundary . . . . 1075.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.6.1 Nature of Solution . . . . . . . . . . . . . . . . . . . . . 1095.6.2 Fixed Energy Ratio Models: r-models . . . . . . . . . . . 110vi6 Two Directed Walks in the Slit . . . . . . . . . . . . . . . . . . . . . 1136.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2 Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . 1206.2.1 Without Interactions . . . . . . . . . . . . . . . . . . . . 1216.2.2 Interacting Model . . . . . . . . . . . . . . . . . . . . . . 1246.3 Solution of Functional Equations . . . . . . . . . . . . . . . . . . 1276.3.1 Without Interactions . . . . . . . . . . . . . . . . . . . . 1276.3.2 With Equal Interactions a = b . . . . . . . . . . . . . . . 1316.3.3 With Interactions, a, b Free . . . . . . . . . . . . . . . . . 1346.4 Exact and Asymptotic Results . . . . . . . . . . . . . . . . . . . 1376.4.1 Case (I) : a = b = 1. . . . . . . . . . . . . . . . . . . . . 1386.4.2 Case (II): a = b = 2. . . . . . . . . . . . . . . . . . . . . 1396.4.3 Case (III): a = 2; b = 1. . . . . . . . . . . . . . . . . . . 1396.4.4 Case (IV): a = b; a < 2. . . . . . . . . . . . . . . . . . . 1406.4.5 Case (V): a = b; a > 2. . . . . . . . . . . . . . . . . . . . 1416.4.6 Case (VI): a < 2; b < 1. . . . . . . . . . . . . . . . . . . 1426.4.7 Case (VII): a > 2; b > 2. . . . . . . . . . . . . . . . . . . 1446.4.8 Case (VIII): a > 2; b < 1. . . . . . . . . . . . . . . . . . 1456.4.9 Case (IX): a < 2; b = 1. . . . . . . . . . . . . . . . . . . 1466.4.10 Case (X): a > 2; b = 1. . . . . . . . . . . . . . . . . . . . 1466.4.11 Case (XI): a = 2; b < 2. . . . . . . . . . . . . . . . . . . 1476.4.12 Case (XII): a > 2; b = 2. . . . . . . . . . . . . . . . . . . 1486.4.13 Case (XIII): ab−a−b = 0. . . . . . . . . . . . . . . . . 1496.4.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.5 Overview and Discussion . . . . . . . . . . . . . . . . . . . . . . 1506.5.1 Infinite Slit Phase Diagram . . . . . . . . . . . . . . . . . 1506.5.2 Force Between the Walls . . . . . . . . . . . . . . . . . . 1536.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 1567 Two Directed Walks in the Slit with Double Interactions . . . . . . . 1577.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.2 Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . 1637.3 Transfer Matrix Approach . . . . . . . . . . . . . . . . . . . . . 166vii7.3.1 Different Widths . . . . . . . . . . . . . . . . . . . . . . 1707.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 1747.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185A Proof of Theorem 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 192viiiList of Tables6.1 The exact value and asymptotic behaviour of the dominant singu-larity when a,b ∈ 1,2 and ab = a+ b. Note that in each case zcdecreases with increasing v. . . . . . . . . . . . . . . . . . . . . 1506.2 The asymptotic behaviour of the dominant singularity when a,b≤2. Again note that in each case, zc is a decreasing function of v andthat zc→ 14 as v→ ∞. . . . . . . . . . . . . . . . . . . . . . . . . 1516.3 The asymptotic behaviour of the dominant singularity when at leastone of a,b > 2. Note that zc decreases with increasing v in all cases. 1516.4 The dominant singularity when b = 1 for the single-walk model. . 152ixList of Figures1.1 A portion of the polymer known as polypropylene. It contains longchains of a repeated monomer known as propene (CH3−CH−CH2). 11.2 A schematic of a linear polymer where each monomer (M) is con-nected to two other monomers, with two terminal monomers. . . . 31.3 Three examples of step sets: (Left) Cardinal Directions, (Center)Steps (1,1) and (1,−1), (Right) Steps (1,1), (1,0) and (1,−1). . 31.4 A walk ϕ of length 34 using the cardinal direction step set. . . . . 41.5 A self-avoiding walk ϕ of length 29 using the cardinal directionstep set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 A directed walk ϕ of length 20 in the lattice Z×Z using the stepset S = {(1,1),(1,−1)}. . . . . . . . . . . . . . . . . . . . . . . 61.7 A directed walk ϕ of length 20 using the step setS = {(1,1),(1,−1)}ending along y = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 A Dyck path of length 20. . . . . . . . . . . . . . . . . . . . . . . 71.9 A bad walk of length 20 with each step inverted after stepping be-low the line y = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 A Motzkin path of length 10. . . . . . . . . . . . . . . . . . . . . 81.11 A schematic of a ring polymer where each monomer (M) is con-nected to exactly two other monomers. . . . . . . . . . . . . . . . 91.12 A self-avoiding polygon of length 44. . . . . . . . . . . . . . . . 91.13 A polygon composed of two directed walks with the same startingand ending vertices. . . . . . . . . . . . . . . . . . . . . . . . . . 10x1.14 We can convert a pair of friendly non-crossing walks into a pairvicious walks by prepending and appending a pair of steps to sep-arate the two walks. . . . . . . . . . . . . . . . . . . . . . . . . . 111.15 The resulting Dyck path using up step sequence of (3,3,1,3,1) anddown step sequence of (1,2,2,3,3). . . . . . . . . . . . . . . . . 122.1 A digraph on three vertices with five directed edges between themwith a size function σ and a weight function w. . . . . . . . . . . 213.1 A Dyck path of length 18 above an non-interactive wall. . . . . . 303.2 An example of a Dyck path prefix ϕ with |ϕ| = 4 and h(ϕ) = 2contributing to z4s2 of D(z,s). . . . . . . . . . . . . . . . . . . . 313.3 Adding steps to existing walks to increase their length (z) by 1 andincreasing/decreasing their height (s) by ±1. . . . . . . . . . . . . 323.4 Forbidden steps taking the walk below y = 0. . . . . . . . . . . . 323.5 A Dyck path of length 18 with interaction parameter a. . . . . . . 343.6 A walk configuration that will lead to a wall interaction. . . . . . . 363.7 Plot of the free energy as a function of interaction parameter. . . . 383.8 The first derivatives of the free energy showing no discontinuitiesat the phase transition. . . . . . . . . . . . . . . . . . . . . . . . 383.9 The second derivatives of the free energy with a discontinuity ata = 2, indicating a second order phase transition. . . . . . . . . . 393.10 When a is small, the model is in a desorbed phase. The walk typ-ically drifts away from the wall and has a zero density of contactsin the limit as n tends to infinity. . . . . . . . . . . . . . . . . . . 413.11 When a is large, the model is in a adsorbed phase. The walk typ-ically stays close to wall and has a positive density of contacts inthe limit as n tends to infinity. . . . . . . . . . . . . . . . . . . . . 413.12 A Dyck path (Left) can be made into a primitive path by prepend-ing and appending steps (Right). . . . . . . . . . . . . . . . . . . 423.13 Partitioning a Dyck path into primitive pieces. . . . . . . . . . . 423.14 Primitive paths have a single wall interaction. . . . . . . . . . . . 43xi3.15 Concatenation of two Dyck paths ϕu with m(ϕu) = 3 and ϕv withm(ϕv)= 4 to form a new Dyck path ϕ ′ of length u+v with m(ϕ ′)=3+4 = 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.16 A dyck path of length 18 with maximal wall contacts. . . . . . . . 503.17 Bounds for κ(α) showing that there is a change in behaviour. . . . 544.1 A Dyck path prefix confined to a slit of width w = 3. . . . . . . . 564.2 Forbidden steps taking the walk above the wall y = w. . . . . . . . 574.3 A Dyck path prefix ϕ of length |ϕ| = 6 confined to a slit of widthw = 3 with interaction parameters a,b. . . . . . . . . . . . . . . . 634.4 Phase diagram of the infinite strip showing the three observed phases. 734.5 Diagram of effective forces experienced by the walls of the slit model. 734.6 Walks of length 18 in a slit of width 6 with maximal wall contactswith top wall and bottom wall. . . . . . . . . . . . . . . . . . . . 765.1 Two directed walks of length 10 of our model that begin and endon the surface. There are two single and two double visits marked.The left-most (start) vertex of the two walks on the wall is notcounted as a double visit. . . . . . . . . . . . . . . . . . . . . . . 815.2 Adding steps to the walks when the walks are away from the wall.There are four possibilities. . . . . . . . . . . . . . . . . . . . . . 835.3 The first boundary term in the functional equation removes the con-tribution from the walks that are produced by appending a SE stepto the bottom walk when its endpoint is on the wall. . . . . . . . . 835.4 The second boundary term in the functional equation removes thecontribution of walks that cross. Such configurations are producedwhen one appends steps to walks that end at the same vertex asshown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.5 Configurations that lead to single (left) and double (right) interac-tion terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85xii5.6 A plot of the probability that a conformation of length 128 has kdoubly-visited sites at a = 1,d = 10.3. This value of d correspondsto the approximate location of the peak in the specific heat at thislength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.7 A plot of the density of a visits calculated at length n = 256. Thishighlights the region where it tends to a non-zero constant and cor-responds well to the regions where za and zd dominate. Note thatfor fixed a and increasing, large d we expect that the density of avisits decreases though remains positive. . . . . . . . . . . . . . . 1045.8 A plot of the density of d visits calculated at length n = 256. Thishighlights the region where it is tends to a non-zero constant andcorresponds well to the region where we have shown that zd is thedominant singularity. . . . . . . . . . . . . . . . . . . . . . . . . 1045.9 A plot of both G(a,a;ρ(a)) and P(a;ρ(a)). The dotted line indi-cates a = 2. In the plot of G(a,a;ρ(a)) we have also marked theasymptotic form computed below. . . . . . . . . . . . . . . . . . 1065.10 The phase diagram of our model. The three phases are as indicatedand the first-order transition is marked with a dashed line, whilethe two second-order transitions are marked with solid lines. Thethree boundaries meet at the point (a,d) = (2,11.55 . . .). . . . . . 1075.11 A plot of the fluctuations in the number of a-visits, ma, for lengthn = 128 as function of a clearly showing two peaks. . . . . . . . . 1116.1 A Dyck path confined between two walls spaced w lattice unitsapart. Each visit to the bottom wall contributes a Boltzmann weighta and each visit to the top wall contributes a Boltzmann weight b.For combinatorial reasons we do not weight the first vertex. . . . . 114xiii6.2 (left) Phase diagram of the infinite strip for a single walk. Thereare three phases: desorbed, adsorbed onto the bottom wall (adsbottom) and adsorbed onto the top (ads top). (right) A diagram ofthe regions of different types of effective force between the wallsof a slit for a single Dyck path. Short range behaviour refers toexponential decay of the force with slit width while long rangerefers to a power law decay. The zero force curve is given by ab =a+ b. On the dashed line there is a singular change of behaviourof the force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.3 Two walks confined between two walls spaced w lattice units apart.Each visit of the bottom walk to the bottom wall contributes aBoltzmann weight a and each visit of the top walk to the top wallcontributes a Boltzmann weight b. For combinatorial reasons wedo not weight the leftmost vertex of either walk. . . . . . . . . . . 1186.4 We form the generating function of all pairs of paths that start inboth surfaces and end anywhere according to their length, and dis-tances of the endpoints from the surfaces. The path depicted con-tributes z9r1s1 to the generating function. . . . . . . . . . . . . . 1206.5 Every pair of paths can be continued by appending directed stepsto their endpoints as shown. While there are at most 4 possiblecombinations, depending on the distance from boundaries, somecombinations will be forbidden. . . . . . . . . . . . . . . . . . . 1226.6 When the endpoints of the walks are close to the boundaries onemust take care to subtract off the contributions of the configurationsthat step outside the strip as depicted here. . . . . . . . . . . . . . 1226.7 (left) When removing the contributions of paths that step outsidethe strip, we over-correct by twice removing those configurationsin which both paths step outside the strip simultaneously. (right)When the endpoints of the paths are close together we must removethe contribution of paths that cross each other. . . . . . . . . . . . 1236.8 Interactions with the boundary are produced when one or bothpaths steps from distance one onto the boundary. . . . . . . . . . 124xiv6.9 The a− b parameter space contains 13 representative points, de-pending on whether a,b = 1, 1 < a,b < 2, a,b = 2, a,b > 2, or ifa = b or if a,b lie on along a special curve ab = a+b. The numbersin this diagram correspond to the cases described in the text. . . . 1386.10 Phase diagram of the infinite strip for the two walk model analysedin this paper. There are four phases: a desorbed phase, a phasewhere the bottom walk is adsorbed onto the bottom wall, a phasewhere the top walk is adsorbed onto the top wall, and a phase whereboth walks are adsorbed onto their respective walls. . . . . . . . . 1536.11 A diagram of the regions of different types of effective force be-tween the walls of a slit. Short range behaviour refers to exponen-tial decay of the force with slit width while long range refers to apower law decay. On full lines there is a change from long to shortrange force decay. On the dashed lines there is a singular changeof behaviour of the magnitude of the force. . . . . . . . . . . . . 1557.1 A self-avoiding walk in an interactive slit with single interactionson both walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.2 (Figure 6, [1]) Force diagram of a self-avoiding walk in a slit withsingle interactions. . . . . . . . . . . . . . . . . . . . . . . . . . 1597.3 A self-avoiding polygon in an interactive slit with double interac-tions on both walls. Wall interactions happen when the walk visitsthe wall and when the walk is one vertex away. . . . . . . . . . . 1607.4 (Figure 15, [1]) Force diagram of a self-avoiding polygon in a slitwith double interactions. . . . . . . . . . . . . . . . . . . . . . . 1617.5 A pair of Dyck paths in a slit of width 4 and length 18. . . . . . . 1617.6 A pair of Dyck paths in a slit with four independent interactions. . 1627.7 A pair of Dyck paths in a slit with double interactions. . . . . . . 1637.8 (Left) The non-crossing condition. (Right) The parity condition. . 1677.9 The no interaction case has weight w = 1 for all possible transitions. 1687.10 Steps in the single interaction case with additional weights w. . . . 1697.11 Steps in the double interaction case with additional weights w. . . 169xv7.12 The value a = a0(w,k) where a slice b = a+ k intersects the zeroforce curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.13 The interval R is the interval of a where Iw(a) and Iw+2(a) have anon-empty intersection. . . . . . . . . . . . . . . . . . . . . . . . 1737.14 A visual representation of the non-negative slices taken (k ≥ 0). . 1747.15 (left) A plot of the zero force point estimates for w = 4 in the a−bplane. (right) The interpolation of the known points to form a zeroforce curve estimate. . . . . . . . . . . . . . . . . . . . . . . . . 1767.16 Plots of the zero force curve estimates for small widths. . . . . . . 1777.17 Plots of the zero force curve for larger widths. . . . . . . . . . . . 1787.18 Possible limiting zero force curves as w→ ∞. . . . . . . . . . . . 1797.19 Location of the zero force points as width increases for large kvalues where they appear to converge. . . . . . . . . . . . . . . . 1807.20 Location of the zero force points as width increases for smaller kvalues where not every point appears to converge. . . . . . . . . . 1807.21 Values a0(w,k) for different k values close to 0 as width increases. 1817.22 Values a0(w,0) as width increases where the points do not appearto converge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817.23 Plot of first differences ∆a0 against inverse width with the line ofbest fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827.24 Plot of a0(w,0) against Logarithm of the width (log(w)) with theline of best fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827.25 A schematic showing the intersection of the zero force curve withthe line b = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.26 Convergence of the zero force point estimate along the line b = 2towards a = 3.30098 . . .. . . . . . . . . . . . . . . . . . . . . . . 184xviAcknowledgementsI would like to show my appreciation and thanks to those who have made this sevenyear adventure one of the most fruitful and enjoyable periods of my life.My sincerest and utmost thanks to my supervisor, Professor Andrew Rech-nitzer, for going above and beyond your supervision duties. Your guidance andsupport in mathematics, teaching, and all things non-academic has made theseyears a very positive experience. I am very thankful to have a supervisor with yourexpertise in your field and the patience and empathy with me as a Ph.D. student.A huge thank you to my supervisory committee, in particular to ProfessorStephanie van Willigenburg, for your encouragement and advice over the years.Your approachability as well as willingness to share your expertise and experi-ences has made the process of seeking assistance so much easier. Your guidancehas always been appreciated over the years! A special thanks to Professor RichardAnstee for your valuable assistance over the years and for agreeing to join the su-pervisory committee with very little prior notification.I would like to thank Professor Aleks Owczarek and Professor Sean Cleary forthe many discussions and their willingness to accommodate all my questions.To the teaching faculty: Professor Mark Mac Lean, Fok-Shuen Leung, andShawn Desaulniers. Thank you for the opportunities and support in helping mehone my teaching skills. It has been a pleasure learning from you and exploring awhole new side to mathematics.To my office mates in MATH 201 and my fellow graduate students whom I hadthe pleasure of working with. Thank you for always being there and for the count-less conversations about mathematics, teaching, and everything under the sun. Anystress I encountered during the project has been greatly diminished knowing that Ixviicould depend on your support.To the support staff in MATH 121. Thank you for your assistance over theyears. Your understanding of the workings of the university is invaluable. Thankyou for taking the extra effort to accommodate my (sometimes very difficult) re-quests. Our many spontaneous conversations will be sorely missed.To the university and external examiners. Thank you for your time and effortin going over this thesis.I would like to acknowledge that this project was funded in part by a Four YearFellowship from the University of British Columbia for which I am thankful for.To complement my academic and professional development, my seven yearsin Vancouver have also been personally fulfilling due to the help and support of agroup of amazing people, not necessarily related to the mathematics department.Without them, this thesis may have been completed slightly earlier, but definitelywith much less fun. In a non-exhaustive, non-sorted list, my deepest personal thankyou:• To Linna, for your time and patience over the years and showing me the bestthat Vancouver has to offer.• To Shannon, for adopting me into your family as well as for the countlessconstructive professional and personal interactions.• To everyone in the sports teams I have the pleasure of partaking in: Critters,Pacman, Real Men, Kaba, Chinomyte, Bumpin’ Grind, and Kiss My Ace formaintaining my mental and physical health.• To my friends in Australia: Howie, Vivian, Sophie, Shun, Dan, Kyri, andJulie Y for their support despite my lack of visits.• To my friends: Kira, Chole, Cheryl, Stala, Heather, Becca, Julie L, Jasmine,Athena, Paul, Shane, Stephanie, May, Mike, and Jenny for putting up withme.Thank you to my parents, without whom I would not be. Words cannot expresshow thankful I am for your continual encouragement and support. Finally, to mybrother. Stay awesome!xviiiChapter 1IntroductionA polymer is a large molecule composed of many connected subunits, known asmonomers [5]. (Figure 1.1). The number and types of connections, or bonds, eachmonomer is allowed to make is based on the chemical properties of the monomers.Suppose we start with n monomers, one natural question to ask is, “How manydifferent ways can these n monomers be connected to form a polymer?”. However,this question is not well defined in the chemistry setting as there are various notionsof “different”. Examples of these include structural isomerism or stereoisomerism[5]. The situation is further complicated if multiple types of monomers are present.Figure 1.1: A portion of the polymer known as polypropylene. It containslong chains of a repeated monomer known as propene (CH3−CH −CH2).To reduce the chemical complexity of the question, we will think of a poly-mer as a single object consisting of a finite number of homogeneous monomers1connected via homogeneous bonds. The homogeneity of the monomers and con-nections allows us to focus on the structural connectivity of the polymer, ratherthan the specific chemical details of each monomer. Under this simplification, theinitial question can be refined to, “How many structurally different polymers canbe created using n homogeneous monomers?”. This also leads to a natural notionof the size of a polymer defined by the number of monomers required to constructit.We will specialise our studies to a special class of polymers known as linearpolymers. These polymers have the property that each monomer is connected toexactly two other monomers, and two terminal monomers are connected to onlyone monomer (Figure 1.2). We can quickly verify that there is exactly one linearpolymer for each length n up to structural isomerism. Hence, the question “howmany structurally different linear polymers of size n?” is not of much interest.However, the physical behaviour of linear polymers such as ribonucleic acid(RNA), deoxyribonucleic acid (DNA), and polypropylene are of interest across arange of scientific and engineering disciplines. In particular, we would like to knowthe number of physical configurations this class of polymers can occupy in physicalspace. From a mathematical view point, we are interested in the number of waysthese polymers can be embedded into space.While the connectivity of the polymer is fixed, a pair of connected monomershas the freedom to move and rotate relative to each other in a continuous man-ner. Hence, embedding a linear polymer into real space (R3) will not provide ameaningful integer number of configurations. Instead, we discretise the space byembedding the polymer to a lattice where we can obtain meaningful integer resultsas well as make use of a range of combinatorial tools. In our case, we will befocusing on the 2-dimension integer lattice (Z2).One mathematical model for studying linear polymer is through the combina-torial class of self-avoiding walks, which will be defined later (see Figure 1.5 forexample). Self-avoiding walks were first studied by Orr [50] and Flory [24] as amodel for linear polymers in a dilute solution. Despite its simple description, itcontains a wealth of interesting open problems. Even simple questions such as“how many self-avoiding walks are there of a given size n?” are extremely difficultfor reasons that will be apparent when we define the class.2Figure 1.2: A schematic of a linear polymer where each monomer (M) isconnected to two other monomers, with two terminal monomers.Let us begin by defining walks in a general setting on the Z2 lattice and thenspecialise to the walks we are interested in. The step setS for a class of walks is acollection of vectors that the walk is allowed to take (Figure 1.3). Each element inthe step set has a well defined size. In the cases we are interested in, each elementin the step set will have unit size. A walk ϕ is a sequence of steps ϕ = s1s2 . . .snwhere each si ∈S . Equivalently, a walk can be defined as the sequence of verticesvisited ϕ = v0v1v2 . . .vn where vi+1− vi ∈ S . In both cases, the size of a walkϕ , denoted |ϕ|, is the sum of the sizes of each step in the walk (Figure 1.4). Forour models, we simply have |ϕ| = n. We adopt the convention that walks start atthe origin v0 = (0,0) to avoid any potential issues with translational symmetries.One subclass of walks we are interested in is the class of non-backtracking walks.This is the class of walks where vi 6= vi+2 for 0 ≤ i ≤ n− 2. We can think of thiscondition as the walk never taking a step in the direction back towards the previousvertex.Figure 1.3: Three examples of step sets: (Left) Cardinal Directions, (Center)Steps (1,1) and (1,−1), (Right) Steps (1,1), (1,0) and (1,−1).A walk ϕ is self-avoiding if it never visits the same vertex twice. That is,vi 6= v j for all i 6= j (Figure 1.5). The class of self-avoiding walks is the set of walks3Figure 1.4: A walk ϕ of length 34 using the cardinal direction step set.with the self-avoiding property. The self-avoiding property naturally captures thevolume exclusion condition for monomers. In other words, two monomers cannotoccupy the same space at the same time. While this is a natural model to study, theanalysis of self-avoiding walk models is very difficult.For example, we can easily verify that number of non-backtracking walks oflength n is given by Wn = 4 ·3n−1 in the square lattice using the cardinal directions.This result is obtained by observing that after the first step, we can extend each ex-isting walk of length n in one of three directions to obtain a new non-backtrackingwalk of length n+1. Hence, we have the limit limn→∞(Wn)1/n = 3.In the identical setting, it has been shown that the analogous limit for self-avoiding walks exists. If we denote the number of self-avoiding walks of size nby cn, then the limit limn→∞(cn)1/n = µ exists and it has been shown that µ ∈(2.62002,2.679192495) [32, 58]. The value µ , known as the growth constant orconnective constant, is representative of the exponential growth in the number ofobjects as size increases. Since µ < 3, this means the set of self-avoiding walksis exponentially rare in the set of all non-backtracking walks. In other words, it isincreasingly difficult to find self-avoiding walks as the size n increases in the set ofall non-backtracking walks (and in the set of all walks).The study of self-avoiding walks is a very difficult task as it is not a Markovianor “memoryless” process. To extend the walk by a single step requires knowledge4of all previous steps. As we increase the size, this process of creating self-avoidingwalks will require infinite memory. As a result, exact enumeration results for self-avoiding walks are rare. Duminil-Copin and Smirnov [19] recently showed thatself-avoiding walks on the hexagon lattice has a growth constant of√2+√2. Cur-rently, no comparable results are known on the square lattice or lattices in higherdimensions. Only numerical estimates are known for certain lattices [31].Figure 1.5: A self-avoiding walk ϕ of length 29 using the cardinal directionstep set.At this point, we turn our attention to a subclass of self-avoiding walks knownas directed walks. In this setting, each element in the step set S has a positivecomponent in the x direction (Center, Right of Figure 1.3). Hence the self-avoidingproperty is naturally satisfied as each x coordinate is visited at most once. Each stepcan be chosen independently creating a Markovian system where we do not haveto consider the previous steps taken. In what follows, we provide the enumerationof some basic directed walk models that will illustrate some characteristics of thewalks we are interested in.As an initial example, we consider the number of walks of length m using thestep set S = {(1,1),(1,−1)}, denote this number Bm (Figure 1.6). For simplicity,we call the step (1,1) the up step and (1,−1) the down step. In this case, each ofthe m steps can be chosen from S independently. This trivially gives the solutionof Bm = 2m walks of length m.Suppose we consider the subset of walks of length m with the same step set S5Figure 1.6: A directed walk ϕ of length 20 in the lattice Z×Z using the stepset S = {(1,1),(1,−1)}.with the added condition that the walk must end along the line y = 0 (Figure 1.7).Let Em denote the number of such walks. If a walk ends along the line y = 0, itmust contain the same number of up steps and down steps. This immediately givesthe restriction that any such walk must have even length m = 2n with n up stepsand n down steps. Hence, not all the steps can be chosen independently. However,we have the freedom to choose the location of the n up steps, or equivalent the ndown steps. Therefore, we see that the number of walks that end on the line y = 0of length m = 2n is given by E2n =(2nn).Figure 1.7: A directed walk ϕ of length 20 using the step set S ={(1,1),(1,−1)} ending along y = 0.In some of the models we are interested in, the line y = 0 acts as an impenetra-6ble wall, or barrier, for the polymer. Hence, we want to further restrict the aboveexample to walks that start at (0,0), end on the line y = 0, and never cross belowit (Figure 1.8). The walks in this model are known as Dyck paths. A completetreatment of Dyck paths as a polymer model will be the focus of Chapter 3.Figure 1.8: A Dyck path of length 20.At this point, we provide a quick argument for enumerating the number ofsuch walks. This enumeration is based on an argument in Andre´ [2] for the ballotproblem. For completeness, the ballot problem involves an election with two can-didates, where one receives strictly more votes than the other during the countingprocess. In the event the two candidates end in a tie, the ballot problem can beexpressed a problem for enumerating Dyck paths.We start with the E2n walks that start and end along the line y = 0 and thensubtract the bad walks that step below the line. For each walk, consider the firsttime that walk steps below the line. We swap each subsequent step: So that each upstep becomes a down step and vice versa (Figure 1.9). The process alters the badwalks to now end at (2n,−2) instead of (2n,0), while leaving Dyck paths invariant.We count the number of walks ending at (2n,−2) by selecting the n+1 down stepsrequired giving( 2nn+1)bad walks. By subtracting the bad walks from the set of allpaths, we get that the number of Dyck paths is counted by the famous CatalannumbersD2n =(2nn)−(2nn+1)= 1n+1(2nn).If instead, we use the step set S = {(1,1),(1,0),(1,−1)} to generate walksthat never cross below the x-axis, we will obtain the set of Motzkin paths thatare counted by the sequence of Motzkin numbers (Figure 1.10). Dyck paths and7Figure 1.9: A bad walk of length 20 with each step inverted after steppingbelow the line y = 0.Motzkin paths are a standard introduction to many enumeration texts and a wealthof literature dedicated to their analysis and behaviour. Further, Dyck path mod-els [10] and Motzkin path models [11] have been used to study the behaviour ofpolymers near an interacting wall. That is, directed walk models maintains theiraccuracy to the physical behaviour of the polymer while being able to produceanalytically tractable results.Figure 1.10: A Motzkin path of length 10.We now turn our attention to the ring polymers. The class of ring polymers(Figure 1.11) is closely related to the class of linear polymers. A ring polymer hasthe property that each monomer is connected with exactly two other monomers,but with no terminal monomers. We can think of a ring polymer as the result ofconnecting the two terminal monomers of a linear polymer.The natural extension of the self-avoiding walk model used to model ring poly-8Figure 1.11: A schematic of a ring polymer where each monomer (M) is con-nected to exactly two other monomers.mers is known as the self-avoiding polygon model. A self-avoiding polygon (Fig-ure 1.12) is a self-avoiding walk that begins and ends at the same vertex (v0 = vn).The same difficulties encountered with self-avoiding walks translate directly to thecase of self-avoiding polygons. Hence, we again turn to directed walk models togain insight into the behaviour of ring polymers.Figure 1.12: A self-avoiding polygon of length 44.Since it is impossible to create loops in directed walk models, we model a ringpolymer by considering two directed walks. We consider two directed walks on astep setS = {(1,1),(1,−1)} that have the same starting and ending vertices (Fig-ure 1.13). This naturally implies both walks must have the same positive integerlength m. Further, we impose friendly non-crossing condition where the two walks9are allowed to share vertices and edges, but their paths may not cross. When weconsider both walks as a single entity, we obtain a loop of length 2m. This friendlycondition is to ensure that walks remains accurate to the physical model. We canconvert a pair of friendly walks into a volume excluding vicious walks that does notshare edges nor vertices by prepending and appending a pair of steps to separatethe two walks (Figure 1.14).Figure 1.13: A polygon composed of two directed walks with the same start-ing and ending vertices.A treatment of two directed walks as a polymer model will be the focus ofChapter 5, Chapter 6, and Chapter 7. In addition to its relevance as polymer mod-els, the two walk models are also of interest and relevance from a mathematicalviewpoint. We provide a quick bijection via Dyck paths to show that two directwalks are also objects enumerated by the Catalan numbers. This bijection is a ver-sion of the bijection provided by Levine [37] that is adapted appropriately. Weillustrate the bijection via the example in Figure 1.13. The non-crossing condi-tion ensures a well defined top path and a bottom path. For each walk, we definea vector for the top walk T and bottom walk B with entries ti (and bi) given bythe number of up steps immediately before the i-th down step. To account for thefull set of possible walks, we append a final entry to T and B for the number of upsteps after the final down step. Since the walks have the share the same starting and10Figure 1.14: We can convert a pair of friendly non-crossing walks into a pairvicious walks by prepending and appending a pair of steps to separatethe two walks.ending vertices, they must contain the same number of up and down steps. Theseensure that T and B are of the same size and have the same sum. The examplein Figure 1.13 gives T = (2,2,0,2,0) and B = (0,1,1,2,2). We increment everyentry in T and B by one to give T+1 = (3,3,1,3,1) and B+1 = (1,2,2,3,3). Wethen complete the bijection by using the new vectors T+1 and B+1 as the up stepand down step sequence, respectively. In this example, the resulting Dyck path willstart with t1 = 3 up steps, followed by b1 = 1 down step, t2 = 3 up steps, b2 = 2down steps, and so on (Figure 1.15). We show that the resulting path is a Dyck pathas follows. Since T and B have the same entry sum, we know the bijection resultsin a walk that ends along the line y = 0. The first step t1 > 0 since we incrementedeach entry by one, hence the first step is always an up step. Finally, we observethat the partial sum of the entries in T must always exceed the partial sum of theentries in B. That is∑iti ≥∑ibi (1.0.1)for all values i. If there exists a value i for which the above inequality was false,the bottom walk would contain more up steps than the top walk at some point,breaching the non-crossing condition. This inequality ensures that the resulting11path does not step below the line y = 0 as there can never be more down steps thanup steps at any point in the path. Hence the resulting path is a Dyck path. Thisbijection shows that pairs of walks are Catalan objects by mapping two walks oflength m to a Dyck path of length 2m+2.Figure 1.15: The resulting Dyck path using up step sequence of (3,3,1,3,1)and down step sequence of (1,2,2,3,3).The exact details of models we study in subsequent chapters will vary and wewill provide a detailed description of the models in each chapter. However, thedirected walk models we consider in this manuscript can be broadly classified as acombination of the following themes.1. Half plane or slit. All the models we will consider involve at least one wall.This defines the boundaries for the walks. In the case of a single wall aty = 0, the walk is restricted to the upper half plane given by Z×Z≥0. If asecond wall is include at a fixed width w, the walks are confined to the slitbetween y = 0 and y = w. Hence, the resulting square lattice is defined byZ×{0,1, . . . ,w}.2. Single walk or multiple walks. In the case of a single walk, the walk takessteps in the set S = {(1,1),(1,−1)} provided that it stays within the con-fines of the boundaries. For multiple walks, we consider cases where bothwalks have the same length. In addition to the boundary conditions, the non-crossing condition means the walks are allowed to touch (share edges andvertices), but not cross.3. Interactions. In the non-interacting models, the walls are inert. There isno benefit or loss for the walks touching the wall. In the interacting case,additional energies are attached to walk when it visits the walls.12The chapters in this manuscript are arranged in a way such that the modelsin each chapter, while self-contained, serve as introduction for the set up and tech-niques used for models in later chapters. Where appropriate, we provide the contextof our models to statistical mechanics of polymers in solution, as well as analogousresults in the self-avoiding walk model.In summary, Chapter 2 contains an introduction to generating functions andtransfer matrices that will be required to build up and analyse the models. A moredetailed treatment of the material can be found in Flajolet and Sedgewick [23]. Weconclude the chapter by building the connection to statistical mechanics via theframework of generating functions.The first model discussed is a single directed walk above a wall. While thismodel has been previously studied [9, 47], we provide a full analysis of both thenon-interacting case as well as the interacting case in Chapter 3. This serves asan illustrative example of the kernel method and related mechanics that will beused for more complicated models in later chapters. For completeness, we alsoinclude an alternative derivation of the model that is commonly used and providessome insight into the behaviour of the two walk model described in Chapter 5.In Section 3.4, we use the single walk model to illustrate the continuity of thefree energy and how it can be extended to a range of models. This will be a keycomponent that will be useful in Chapter 7.We extend the model in Chapter 3 to include a second wall. This model waspreviously studied by DiMarzio and Rubin [17] using a transfer matrix method andby Brak et al. [10] using an iterative method on the width of the slit. In Chapter 4,we reformulate the model in a way such that it is amenable to the kernel method.The results we obtain are consistent with those obtained by Brak et al. [10].Chapter 5 is the first of the new models that we studied. It contains two directedwalks above an interacting wall. We construct the necessary functional equationsrequired to apply the kernel method, which includes an additional step of extractinga “constant term solution”. We provide the analogous analysis of the model to thatof the single walk in Chapter 3.As with the single walk case, a natural extension of the model is to furtherconfine the walks to a slit using a second wall. In Chapter 6, we discuss the modelwith only single interactions present. We use the kernel method to extract the13location of the singularities which allows us to determine asymptotic behaviour ofthe system.Finally, we consider the full directed walks in a slit model by incorporatingdouble interactions. This is the topic of Chapter 7. In this model, we are unableto find enough symmetries in the functional equations to extract the solutions orsingularities. Instead, we turn to a transfer matrix approach where we are able todetermine highly precise numerical estimates for the zero force curve of the system.14Chapter 2PreliminariesThis chapter covers the standard definitions and some basic results from enumer-ation and transfer matrices that will be used in later chapters. It also provides theconnection between enumeration and statistical mechanics that will be used laterin the manuscript. The reader is invited to proceed to Chapter 3 if they are alreadyfamiliar with this content.2.1 Generating FunctionsGenerating functions are the primary mathematical object used when storing andanalysing information about the models we are interested in. The definitions andresults from this section will be largely borrowed from Flajolet and Sedgewick[23].Definition 2.1. A combinatorial class A is a finite or countable set with a sizefunction such that:1. the size of any element is a non-negative integer;2. the number of elements of any given size is finite.Equivalently, a combinatorial class can be defined as a pair (A , | · |) such thatA is countable and | · | : A 7→ N such that the inverse image of any non-negativenumber is finite.15If A is a combinatorial class, the size of an element a ∈A is denoted |a|. Fora combinatorial class A , the number of elements of size n is denoted An.Example 2.2. The set of Dyck paths forms a combinatorial classD . For each pathϕ ∈D , the size of ϕ is the number of steps in the walk. Further, since we know thenumber of walks for any even number m = 2n is given by the Catalan numbers, wehave D2n = 1n+1(2nn).Definition 2.3. Let A be a combinatorial class. The counting sequence of A isthe sequence of integers {An}n≥0.Definition 2.4. The ordinary generating function of a combinatorial classA is theformal power seriesA(z) =∞∑n=0Anzn.Definition 2.5. Given a formal power series f (z) = ∑ fnzn, the coefficient of zn isdenoted [zn] f (z). That is[zn] f (z) = [zn](∞∑n=0fnzn)= fn.In the case where n = 0, we have a alternative method of extracting the coeffi-cient [z0] by f (0) = [z0] f (z) sincef (0) =(∞∑n=0fn(0)n)= f0 = [z0] f (z).Example 2.6. Every Dyck path must be of even length. Hence, Dm = 0 for m odd.For even integers m = 2n, the ordinary generating function for Dyck paths D isgiven by the power seriesD(z) =∞∑n=0D2nz2n =∞∑n=01n+1(2nn)z2n. (2.1.1)Similarly, for any even m = 2n, we haveD2n = [z2n]D(z) =1n+1(2nn). (2.1.2)16An idea that will be used extensively in later chapters is the concept of a mul-tivariable generating function.Definition 2.7. Let {An,k}n≥0,k≥0 be a sequence of numbers depending on two in-teger valued indices n and k. The bivariate generating function of such a sequence{An,k} is defined asA(z,u) =∑n,kAn,kukzn. (2.1.3)For a combinatorial class A , the number An,k is the number of objects ϕ ∈A such that |ϕ|A = n and some other parameter χ(ϕ) = k. As for the singlevariable case, the variable z is conjugate to the size of the object and the variable uis conjugate to the parameter χ . This process can be extended to multiple variables.Definition 2.8. Consider a combinatorial classA . A multidimensional parameterχ = (χ1,χ2, . . . ,χd) on the class is a function from A to the set Nd of d-tuplesof natural numbers. The counting sequence of A with respect to the size andparameter χ is defined byAn,k1,k2,...kd = |{a ∈A | |a|= n,χ1(a) = k1,χ2(a) = k2, . . . ,χd(a) = kd ,}| .(2.1.4)The multi-index notation can be used to simplify notation where appropriate.Let u = (u1,u2, . . . ,ud) be a vector of formal variables and k = (k1,k2, . . .kd) be ainteger vector of the same dimension, then uk is defined asuk = uk11 uk22 · · ·ukdd . (2.1.5)With this notation, we can make the following definition.Definition 2.9. Let {An,k} be a multi-index sequence of numbers where k ∈ Nd .The multivariate ordinary generating function of the sequence {An,k} is the formalpower seriesA(z,u) =∑n,kAn,kukzn. (2.1.6)17Given a class A and a multidimensional parameter χ , we can convert the gen-erating function into a summation of its elements asA(z,u) = ∑a∈Auχ(a)z|a|. (2.1.7)The following is a summary of the results that will be required in this thesis.More details can be found in Flajolet and Sedgewick [23], which provides a wealthof information and results related to the asymptotic behaviour of generating func-tions.Theorem 2.10 (Pringsheim’s Theorem). (Flajolet and Sedgewick [23], page 240.)If f (z) is representable at the origin by a series expansion that has non-negativecoefficients and radius of convergence Rz, then the point zc = Rz is a singularity off (z).In particular, Pringsheim’s Theorem holds for all ordinary generating func-tions. The point zc = Rz is known as a dominant singularity of f (z) as it is asingularity of smallest magnitude (I.e. closest to the origin).Definition 2.11. A number sequence {an}n≥0 is said to be of exponential order Kn,denoted an ./ Kn, if and only if limsup |an|1/n = K.This gives and upper and lower bound on the sequence {an}. Given ε > 0,1. the number |an| exceeds (K− ε)n infinitely often, and2. the number |an| is dominated by (K + ε)n except for finitely many values ofn.The value K is referred to as a growth rate of the sequence. If the sequence comesfrom a generating function f (z), then K is said to be the growth rate of the function.Theorem 2.12. (Flajolet and Sedgewick [23], page 240.) If f (z) is analytic at 0and Rz is the modulus of a singularity nearest to the origin, then the coefficientfn = [zn] f (z) satisfiesfn ./(1Rz)n.18These results are stated here for the single variable (univariate) case but can beextended to the multivariate case. In the case f (z,u), the coefficients [zn] f (z,u) =fn(u), which means the radius of convergence is a function of the parameters Rz =Rz(u).The nature of the function’s singularities provides information about the asymp-totic behaviour of the coefficients. This is particularly helpful in the cases wherewe were not able to determine the coefficient in any explicit form. We decomposethe main result into two steps. The first step consists of a rescaling[zn] f (z) = ρ−n[zn] f (ρz). (2.1.8)This allows us to rescale any generating function so that it is singular at z = 1. Westate the result for the asymptotic form of the coefficients provided in Flajolet andSedgewick [23].Theorem 2.13 (Theorem VI.1, Flajolet and Sedgewick [23]). Let α be an arbitrarycomplex number in C\Z≤0. The coefficient of zn inf (z) = (1− z)−α (2.1.9)admits for large n a complete asymptotic expansion in descending powers of n,[zn] f (z)∼ nα−1Γ(α)(1+∞∑k=1eknk)(2.1.10)where ek is a polynomial in α of degree 2k and Γ is the well known Gamma func-tion.Theorem 2.13 can be extended to functions of the form (1− z)−α(1z log11−z).See Theorem VI.2, Flajolet and Sedgewick [23]. A full analysis is detailed inSection VI.9 and VI.10 of Flajolet and Sedgewick [23].The last result in this section is not specific to generating functions but ratherto complex functions in general.Theorem 2.14 (Lalı´n and Smyth [35]). Let h(z) be a non-zero complex polynomialof degree n having all its zeros in the closed unit disc |z| ≤ 1. Then for d > n and19any λ on the unit circle, the self-inverse polynomialP(λ )(z) = zd−nh(z)+λ znh¯(1z)(2.1.11)has all its zeros on the unit circle.2.2 Transfer MatricesThis section contains a summary of the basic graph theory required to develop thetransfer matrix method required for our model in Chapter 7.Definition 2.15. A directed graph (or digraph) G is the pair (V,E) of its vertex setV and its edge set E ⊆V ×V . Self-loops, edges of the form (v,v), are allowed.For a digraph G with vertex set identified to the set {1,2, . . . ,n}, we weighteach edge e = (a,b) with a formal indeterminate quantity ga,b for which we allowthe substitution of positive weight values.Definition 2.16. The weighted adjacency matrix of a weighted digraph G is thematrix A with entries determined by:A[i, j] =gi, j if (i, j) ∈ E0 otherwise.(2.2.1)This process can be reversed. Given a matrix A of size n×n with non-negativeentries, we can construct an asssociated graph G with vertex set {1,2, . . . ,n} andedge set containing (i, j) of weight gi, j = A[i, j] > 0. No edge is present if theweight is 0.Definition 2.17. A non-negative square matrix A is called irreducible if its associ-ated graph is strongly connected. That is, there exists a directed path between anytwo vertices.Definition 2.18. A strongly connected direct graph G is said to be periodic withparameter d if and only if the vertex set can be partitioned into d classes V =V0 ∪V1 ∪ ·· · ∪Vd−1 in such a way that any edge (a,b) with a ∈ Vi must have b ∈20Vi+1 mod d . The largest possible d is called the period. If no decomposition existswith d ≥ 2, then the graph is called aperiodic.A non-negative square matrix A is said to be aperiodic if its associated graph isaperiodic. Similar, A is said to have period d if its associated graph has period d.A transfer matrix is a simple extension of the digraph model where each edgeis weighted, with possibly different weights.Definition 2.19. Given a digraph G = (V,E), a size function is any function σ :E 7→ Z≥1. A sized graph Γ is a pair (G,σ), where σ is a size function.For a sized graph Γ and a weight function w : E 7→ R+, we can associate atransfer matrix T (z) entry-wise byT (z)[i, j] = ∑e∈Edge(i, j)w(e)zσ(e), (2.2.2)where Edge(i, j) is the set of edges connecting i to j.Example 2.20. Figure 2.1 shows an example of a digraph G = (V,E) on threevertices V = {1,2,3} and five edges E = {a,b,c,d,e} with a size function σ and aweight function w.Figure 2.1: A digraph on three vertices with five directed edges between themwith a size function σ and a weight function w.We can associate a transfer matrix entry-wise. The size function σ determinesthe exponent of variable z and the weight function determines the coefficient of21each term. The associated transfer matrix for the given digraph isT (z) =z2 2z1 03z2 0 4z1 +5z20 0 0 . (2.2.3)Note that edges d and e both contribute to entry T (z)[2,3]. These contributions arecombined additively.Lemma 2.21. Given a sized graph with associated transfer matrix T (z), the ordi-nary generating function F<i, j>(z) of the set of paths from i to j, where z countsthe size, is the entry i, j of the matrix (I−T (z))−1, where I is the identity matrix.F<i, j>(z) =((I−T (z))−1)[i, j]. (2.2.4)Lemma 2.22. (Flajolet and Sedgewick [23], page 343.) Let T (z) be an irreducibletransfer matrix. Then all entriesF<i, j>(z) =((I−T (z))−1)[i, j] (2.2.5)have the same radius of convergence in the parameter z, Rz, which can be definedin two equivalent ways:1. as Rz = λ−1 with λ the largest positive eigenvalue of T (z),2. as the smallest positive root of the determinantal equation: det(I−T (z)) =0.Furthermore,the point Rz is a simple pole of each F<i, j>(z).If T (z) is irreducible and aperiodic, then zc = Rz is the unique dominant singu-larity of each F<i, j>(z) and for computable constants φi, j > 0 and 0 ≤ Λ < λ wehave[zn]F<i, j>(z) = φi, jλ n +O(Λn). (2.2.6)This is equivalent to saying that for any i, j in a irreducible and aperiodic trans-fer matrix, the coefficient [zn]F<i, j>(z) ./ λ .Lemma 2.23. The transfer matrix T (z) and its transpose have the same spectrum.22Proof. We can show that the two matrices have the same characteristic function:det(T (z)−λ I) = det(T (z)−λ I)′ = det(T (z)′−λ I). (2.2.7)The first equality results from the invariance of determinant under transposition.The second equality results from the matrices being identical.In the models we are interested in, each edge of the sized graph has a constantsize of 1. Hence, the transfer matrix T (z) = zA for some adjacency matrix A withnon-negative entries.Theorem 2.24 (Perron-Frobenius Theorem). Let A be a matrix with non-negativeentries that is assumed to be irreducible. The eigenvalues of A can be ordered insuch a way thatλ = |λ2|= . . .= |λd |> |λd+1| ≥ |λd+2| ≥ . . . , (2.2.8)and all the eigenvalues of largest modulus are simple. Furthermore, the quantityd is precisely equal to the period of the associated graph. In particular, in theaperiodic case d = 1, there is a unicity of the dominant eigenvalue.One intermediate step in the proof the Perron-Frobenius Theorem is the fol-lowing, which we will require explicitly for an upcoming result.Corollary 2.25. The matrix T (z) = zA has positive left and right eigenvectorsassociated to the dominant eigenvalue.The generating functions obtained from these results can be expanded to themultivariate case by altering the weight function of the associated transfer ma-trix (Equation 2.2.2). Explicitly, we now define a weight function w(e) = uk forsome multi-index monomial (Equation 2.1.5), giving a multivariate transfer matrixT (z,u). Following this through, we get the generating function of paths from i toj is given by F<i, j>(z,u), which grows like the dominant eigenvalue λ (u). Thetransfer matrix T (z,u) = zA(u) for some adjacency matrix A(u) where the entriesare non-negative coefficients in the parameters u.In the single variable case, or in the case where we specify numerical valuesfor u, we can determine numerical bounds for the dominant eigenvalue.23Definition 2.26. Given a finite adjacency matrix A of size N and a positive vectorx of size N, let vi(x) = (Axx)ixi . Define the minimum and maximum of such ratios tobem(x) = minivi(x), M(x) = maxivi(x). (2.2.9)The following theorem of Collatz [13] regarding spectral radius holds in amuch more general setting in linear algebra. However, by Pringsheim Theorem(Theorem 2.10) for non-negative and irreducible matrices, we can refine the theo-rem to an interval in the non-negative real numbers.Theorem 2.27 (Collatz [13]). Given a positive vector x > 0, the dominant eigen-value of an irreducible and nonnegative adjacency matrix A, denoted λ (A), is inthe intervalλ (A) ∈ (m(x),M(x)) , (2.2.10)or in the case where x is a positive dominant eigenvector of A,m(x) = λ (A) = M(x). (2.2.11)Proof. Let λ be the dominant eigenvalue of A and v be the corresponding lefteigenvector (Corollary 2.25). Then we have vA = λv. For a positive vector x, wehavevA = λv (2.2.12)v(Ax) = λv ·x. (2.2.13)24Denote the vector z = Ax∑ivizi =∑iλvixi (2.2.14)∑ivi (λxi− zi) = 0 (2.2.15)∑ivixi(zixi−λ)= 0. (2.2.16)In the case where zixi −λ = 0 for every i, we have x= v and we recover the dominanteigenvector. Otherwise, there must be at least one i where zixi −λ 6= 0. Since v andx are non-negative, we have vixi > 0 for all i. This implies the existence of indicesk, l such thatzkxk−λ > 0 > zlxl−λ (2.2.17)zkxk> λ > zlxl. (2.2.18)Hence the dominant eigenvalue must be bounded between the largest and smallestof zixi over the range of possible i’s.2.3 Connection to Statistical MechanicsStatistical mechanics is the field of study relating the microscopic states of a sys-tem to its macroscopic properties. This section provides an brief overview of thestatistical mechanics of polymer models in the framework of generating functions.We refer the reader to Janse van Rensburg [47] for a full treatment of the topic aswell as details for the statistical mechanics of related models.In our models, we will be looking at polymer behaviour using combinatorialclasses of directed walks. The details of each combinatorial class will vary accord-ing to the polymer model we study. For notation purposes, we denote S to be thecombinatorial class of directed walks we use to describe the polymer model. Theclasses S we are interested in will be a combination of the following:1. A single, or multiple Dyck paths;252. contained in the upper half plane, or a in a slit of finite width w;3. with or without interactions.For each class, we define a size for each ϕ ∈S , denoted |ϕ|, usually countedby the number of steps in a ϕ . Further, we will define a multidimensional parameterχ as in Definition 2.8 to represent the interactions that the model experiences. Themultivariate generating function S(z,u) is the grand canonical partition function.S(z,u) = ∑ϕ∈Suχ(ϕ)z|ϕ|. (2.3.1)Given a non-negative integer n, let Sn ⊂S be the set of elements of size n. Wedefine the partition function to beZn(u) = ∑ϕ∈Snuχ(ϕ). (2.3.2)The partition function is the restriction of the grand canonical partition function toa system of a fixed size n. We can rewrite the grand canonical partition function asS(z,u) = ∑n≥0Zn(u)zn, (2.3.3)and equivalentlyZn(u) = [zn]S(z,u). (2.3.4)The partition function is the natural object of study from a statistical mechanicsperspective. However for the methods we will use to study the models, the grandcanonical partition function is the more natural object to study.In the context of thermodynamics, each ϕ ∈S represents a microstate of thesystem with total energy Eϕ obtained from the sum of a finite number of energycontributions. Suppose that the system has d different sources of energy contribu-tion, thenEϕ =d∑i=0mi(ϕ)(−εi) , (2.3.5)where−εi is the energy contribution from i-th contribution and mi(ϕ) is the numberof times the i-th contribution occurs in ϕ . In this context, the partition function is26defined on the set of microstates of a fixed size (Sn) asZn = ∑ϕ∈Snexp[−EϕkBT], (2.3.6)where kB is the Boltzmann constant and T is the temperature of the system. Byrewriting, we getZn = ∑ϕ∈Snexp[−EϕkBT](2.3.7a)= ∑ϕ∈Snexp[∑di=0 mi(ϕ)εikBT](2.3.7b)= ∑ϕ∈Snd∏i=0exp[εikBT]mi(ϕ). (2.3.7c)Comparing the above with Equation 2.3.2 gives a thermodynamic interpreta-tion of the variables u = (u1,u2, . . .ud) as ui = exp[εikBT], which is known as theBoltzmann weight of the i-th interaction. The parameter χi(ϕ) = mi(ϕ) is thenumber of times the interaction occurs in microstate ϕ . That is, each ui encodesinformation about the strength of the different energy contributions. Depending onthe context, it is useful to define µi = εikBT so that we can write the variables ui = eµiin an exponential form.From the grand canonical partition function, which provides information aboutthe microstates of the system, we can extract global or macroscopic informationabout the system by studying its asymptotic behaviour as the size n goes to infinity.The limiting free energy is one example of a macroscopic property that resultsfrom the asymptotic behaviour of the microstates. When the context is clear, thelimiting free energy is referred to as the free energy.Definition 2.28. Given a model with grand canonical function S(z,u)=∑n≥0 Zn(u)zn,the free energy is defined asκ(u) = limn→∞1nlog(Zn(u)) . (2.3.8)Let zc(u) be the dominant singularity in S(z,u). We can rewrite the free energy27asκ(u) =− log(zc(u)) . (2.3.9)The reduces the problem of determining the free energy of a model to findingthe location of the dominant singularity in the grand canonical partition function.A phase transition is a discontinuity in the free energy κ and indicative of a changein behaviour of the system. The phase transition is said to be of p-th order for thesmallest value p where the p-th derivative is discontinuous.We can investigate the system further by looking at the average number ofinteractions, 〈mi〉, for each energy εi for the system at a fixed size n〈mi〉n =∂∂uilog(Zn(u)) =∂∂ui Zn(u)Zn(u). (2.3.10)Taking the limit as n→∞ results in a density on the number interactions in thesystem of a given typelimn→∞〈mi〉nn= limn→∞1n∂∂uilog(Zn(u)) (2.3.11)= ∂∂uiκ(u). (2.3.12)This final equation gives an alternative method of determining the density ofa particular interaction provided we have information regarding the free energy ofthe system.28Chapter 3Directed Walk Above a WallThis chapter is a summary of known results related to directed walk models of apolymer in a presence of a sticky wall. Chapter 4 deals with the case of multiplewalls. Combined, these two chapters aim to serve two purposes. Firstly, it moti-vates the models we study in later chapters. Secondly, it provides an introductionto the kernel method via illustrative examples. The kernel method is a tool that willbe used extensively for the models in later chapters. A more detailed overview ofthe technique can be found in [7]. A detailed analysis of the statistical mechanicsrelated to interacting walks and similar models is covered in Janse van Rensburg[47].In Section 3.1, we consider directed walks in the upper half plane that startand end along the line y = 0 (Dyck paths). While the results can be found in mostintroductory texts to enumerative combinatorics, we approached this model in away that can be naturally extended to incorporate interactions, a second walk, orin a slit setting. The Dyck path model will be extended to incorporate interactionswith the wall in Section 3.2.3.1 Non-Interacting ModelThe simplest model we consider is a single directed walk above a single wall,taking steps (1,1) (up step) or (1,−1) (down step). The walk may touch y = 0 butnot go below. A walk ϕ that start at (0,0) and ends along the x-axis at some point29(2n,0) is called a Dyck path of length |ϕ|= 2n (Figure 3.1). The set of Dyck pathsD refers to the collection of all such walks of finite length. This model correspondsto a single polymer in dilute solution that does not interact with the surface.Figure 3.1: A Dyck path of length 18 above an non-interactive wall.This gives rise to the associated ordinary generating function D(z) where zmarks the length of the walk. To determine an explicit function for D(z), we beginby looking at a larger class of walks. The set of Dyck path prefixes D∗ is thecollection of walks satisfying the same criteria as a Dyck path, but without therestriction that it has to end on the x-axis (Figure 3.2).We want to retain information about the length of each walk in this larger classas well as record information about the height of the final vertex. We use z to markthe length of each walk as per the Dyck path case and introduce a new variable s torecord the height of the final vertex. This results in a bivariate generating function(Definition 2.7) in the variables z and s.Definition 3.1. For ϕ ∈ D∗, let h(ϕ) denote the height of the final vertex in thewalk ϕ . The associated bivariate generating function is given byD(z,s) = ∑ϕ∈D∗z|ϕ|sh(ϕ). (3.1.1)We establish a functional equation satisfied by D(z,s) using a column by col-umn construction. That is, we obtain larger elements in the set by appending fea-sible steps to existing elements. The resulting functional equation isD(z,s) = 1+ z(s+ 1s)·D(z,s)− zs· [s0]D(z,s). (3.1.2)30Figure 3.2: An example of a Dyck path prefix ϕ with |ϕ| = 4 and h(ϕ) = 2contributing to z4s2 of D(z,s).The left hand side represents all possible walks in the set. We explain each of theterms on the right hand side of the equation:1. The trivial walk of length 0 contributes the initial 1 in the functional equa-tion.2. Every walk can be extended by appending an up step (1,1) or down step(1,−1) to an existing walk, contributing zsD(z,s) and zs−1D(z,s) respec-tively (Figure 3.3).3. Appending a down step for walks ending at height 0 will result in the walkstepping below the wall y = 0 (Figure 3.4). We account for this boundarycondition by subtracting off the contributions of the down step to walks atheight 0. This is given by zs−1 · [s0]D(z,s).For a integer i ≥ 0, we will simplify notation by denoting Di(z) = [si]D(z,s).For example, the set of Dyck paths is counted by the set of walks where the heightof the final vertex is 0. Hence we have D(z) = [s0]D(z,s) = D0(z).Using this notation, we rearrange Equation 3.1.2 to obtain[1− z(s+ 1s)]·D(z,s) = 1− zs·D0(z). (3.1.3)31Figure 3.3: Adding steps to existing walks to increase their length (z) by 1and increasing/decreasing their height (s) by ±1.Figure 3.4: Forbidden steps taking the walk below y = 0.The factor on the left hand side is known as the kernel K(z,s) and will play animportant role in future calculationsK(z,s) = 1− z(s+ 1s). (3.1.4)In this notation, Equation 3.1.3 becomesK(z,s) ·D(z,s) = 1− zs·D0(z). (3.1.5)At this point, we aim to simplify the left hand side by looking for a solutionsˆ = s(z) such that K(z, sˆ) = 0. Provided that our choice of sˆ results in D(z, sˆ)32being convergent as a formal power series in z, we can eliminate the left hand sidecompletely to get0 = 1− zsˆ·D0(z). (3.1.6)We look for a valid choice of sˆ by considering the possible solutions for sˆ.Solving for sˆ, we get two possible solutions for sˆsˆ1 =1−√1−4z22z= z+O(z3), (3.1.7a)sˆ2 =1+√1−4z22z= z−1− z+O(z3). (3.1.7b)A walk of length n can have a final vertex of at most height n. In fact, thereis an unique walk that accomplishes this consisting of n up steps. Hence, for eachchoice of k≥ 0 we have [zk]D(z;s) = Pk(s) is a polynomial in s of degree k. Makingthe substitution the two different sˆ, we seePk(sˆ1) = Constant+O(z), (3.1.8)Pk(sˆ2) = z−k +O(z−n+1). (3.1.9)Making these substitutions into D(z;s), we obtain, through a slight abuse ofnotation,D(z; sˆ1) =∞∑k=0zkPk(sˆ1) =∞∑k=0Constant · zk +O(zk+1), (3.1.10)D(z; sˆ2) =∞∑k=0zkPk(sˆ2) =∞∑k=01+O(z) . (3.1.11)We see the substitution D(z; sˆ2) does not produce a valid generating function.If we rearrange the partial sum D(z; sˆ2) by collecting terms by powers z, we getthat the partial sumsn∑k=0zkPk(sˆ2) = n+O(z), (3.1.12)which is divergent in the limit as n→ ∞. In other words, the substitution of s = sˆ2into D(z;s) does not converge in the ring of formal power series in z with integercoefficients and is an invalid substitution.33In the other case, each series Pn(sˆ1) starts with a constant term. Hence, eachterm znP(sˆ1) = O(zn) for the series D(z; sˆ1). That is, each coefficient [zn]D(z; sˆ1)consists of positive contributions from a finite number of P(sˆ1). Hence, the seriesD(z; sˆ1) converges in the ring of formal power series in z with integer coefficients.Since s = sˆ1 is a valid substitution, we can eliminate the left hand side com-pletely as K(z, sˆ1) = 0. This leaves a single unknown function D0(z) on the righthand side, which we rearrange to giveD0(z) =sˆz. (3.1.13)Rewriting sˆ in terms of z, we recover the well known Catalan generating func-tionD0(z) = D(z) =1−√1−4z22z2. (3.1.14)For completeness, the dominant singularity for D(z) is due to the discontinuityof the square root at z = 12 . Therefore, the free energy of this system is κ = log(2).3.2 Interacting ModelOne natural extension of the model is to consider the interaction between the walland the polymer. Starting with the set of Dyck path prefixes D∗, we add an energy−εa for each visit of the walk to the wall y = 0, giving it a Boltzmann weight ofa = exp[εakbT](Figure 3.5). To reduce the complexity of computations, we omit theweight of the initial contact at (0,0). The results can be easily modified to includethis weight if required.Figure 3.5: A Dyck path of length 18 with interaction parameter a.34Definition 3.2. The associated generating function for this model is given byD(z,s;a) = ∑ϕ∈D∗z|ϕ|sh(ϕ)am(ϕ), (3.2.1)where m(ϕ) is the number of times the walk ϕ touches the wall y = 0.Consider the set of all Dyck path prefixes of length n, denotedD∗n , We have thepartition functionZn(a) = ∑ϕ∈D∗nexp[εam(ϕ)kbT]= ∑ϕ∈D∗nam(ϕ). (3.2.2)We establish a functional equation satisfied by D(z,s;a) using the analogouscolumn by column construction as per the non-interacting modelD(z,s;a) = 1+ z(s+ 1s)·D(z,s;a)− zs· [s0]D(z,s;a)+ z(a−1) · [s1]D(z,s;a).(3.2.3)Analogous to the previous case, the left hand side accounts for all possiblewalks with interactions. The first three terms on right hand side follow mutatismutandi from the non-interacting model (Equation 3.1.2).The final term of the right hand side results when we account for the updatedweight of a walk when it touches the wall. Every walk that touches the wall muststep from height 1 down to height 0. We mark these walks interaction parametera and subtract the non-interacting versions of those exact walks from the model.This givesz(a−1) · [s1]D(z,s;a). (3.2.4)We now proceed as per the previous model by rearranging to get the kernelK(z,s) ·D(z,s;a) = 1− zs·D0(z;a)+ z(a−1) ·D1(z;a). (3.2.5)At this point, the right hand side of the function contains two unknown functions.We want to find an equation analogous to Equation 3.1.5 in the sense that the righthand side only depends on a single unknown function. To obtain this, take thecoefficient [s0] of both sides of the equation to get a relation between D0(z;a) and35Figure 3.6: A walk configuration that will lead to a wall interaction.D1(z;a) that we can use to eliminate the latter. The relation we get isD0(z;a) = 1+ zaD1(z;a). (3.2.6)Combinatorially, this is equivalent to the observation that every walk ending atheight 0 is either the trivial walk or obtained by appending a step to a walk endingat height 1. Combining this information, we eliminate D1(z;a) from Equation 3.2.5to getK(z,s) ·D(z,s;a) = 1a+(a−1a− zs)·D0(z;a). (3.2.7)The kernel of this functional equation is identical to the non-interacting model.Therefore, the solutions sˆ with K(z, sˆ)= 0 are identical to the non-interacting model(Equation 3.1.7a and Equation 3.1.7b).We use the same argument on the coefficients [zn]D(z,s;a) as per the non-interacting model to get that s = sˆ2 results in an invalid substitution as D(z, sˆ2;a)contains divergent partial sums. The same argument is used to show that the sub-stitutions = sˆ1 =1−√1−4z22z(3.2.8)into D(z, sˆ1;a) gives convergent power series in z and a.36Hence, we can substitute s = sˆ1 into Equation 3.2.7 to give0 = 1a+(a−1a− zsˆ)·D0(z;a). (3.2.9)Solving for D0(z;a), we getD0(z;a) =11−a(1− zsˆ) . (3.2.10)Rewriting as a function of z, we haveD0(z;a) =11−a(1−√1−4z22) . (3.2.11)Note that D0(z;a) contains two sources of singularities.1. The square root in the denominator gives a singularity at z = 12 .2. The zeros of the denominator gives the second singularity. Solving the de-nominator for z, we get that singularity at z =√a−1a .Comparing the two sources, we see that the dominant singularity zc is given byzc(a) =12 a≤ 2√a−1a a > 2. (3.2.12)This gives a free energyκ(a) =log(2) a≤ 2log(a√a−1)a > 2. (3.2.13)At a = 2, this model undergoes a phase transition (Figure 3.7). Looking at thederivatives of the free energy κ , we can validate that the free energy is continuous inits first derivative (Figure 3.8) and has a second order phase transition (Figure 3.9)We can obtain information about each of the phases by looking at the averagenumber of interactions between the walk and the wall by using Equation 2.3.10.37Figure 3.7: Plot of the free energy as a function of interaction parameter.Figure 3.8: The first derivatives of the free energy showing no discontinuitiesat the phase transition.The behaviour of the system in each of the phases is governed by the behaviourof the dominant singularity. Hence, we need to partition the system into threecases based on the location and nature of the singularity. Section VI.9 of Flajo-let and Sedgewick [23] provides the details of the three cases of this compositionschema. Broadly speaking, we determined the asymptotic expansion of the co-efficients in the numerator and denominator of Equation 2.3.10 separately. Weaccomplished this by taking a series expansion around the dominant singularity ofEquation 3.2.11 and then extracted the leading term in the asymptotic expansiondue to the largest non-analytic term in the series.38Figure 3.9: The second derivatives of the free energy with a discontinuity ata = 2, indicating a second order phase transition.• In the case where a < 2, we are in the subcritical case where the dominantsingularity zc(a) = 12 is driven by the square root factor. The numerator ofEquation 2.3.10 is given by the uniform asymptotic expansion,a∂∂aD(z;a) ∼z→ 122a(a−2)2 +2a(a+2)√2(a−2)3 ·√1−2z+O(1−2z). (3.2.14)The dominant term in the asymptotic expansion of the coefficients is givenby the non-analytic square root term, which we can expand to give[zn]a ∂∂aD(z;a)∼a(a+2)√2√pi(2−a)3 ·2n ·(n−32 + 38n−52 +O(n−72)). (3.2.15)We apply the same technique to the denominator to obtain an asymptoticexpansion of the coefficients[zn]D(z;a)∼ a√2√pi(2−a)2 ·2n ·(n−32 + 38n−52 +O(n−72)). (3.2.16)By taking the ratio of the asymptotic form for the numerator and denomina-39tor, we get that for a < 2.〈ma〉n =a+22−a +O(n−1). (3.2.17)• The supercritical case consists of a > 2. The dominant singularity, zc(a) =√a−1a is driven by the denominator of D0(z;a). We considered the expansionaround z = zc(a), and via the mechanics as for the previous case, we get[zn]a ∂∂aD(z;a)∼(a−2)24(a−1)2 ·(a√a−1)n· (n+1) . (3.2.18)The denominator in this case becomes[zn]D(z;a)∼ (a−2)22(a−1) ·(a√a−1)n. (3.2.19)Asymptotically, we have for a > 2,〈ma〉n =a−22(a−1) (n+O(1)) . (3.2.20)• In the case where a = 2, we are in the critical case. We determine the asymp-totic form for the mean number of contacts by applying Equation 2.3.10 andthen making the substitution a = 2. The asymptotic expansion of the numer-ator is[zn]a ∂∂aD(z;a)∣∣∣a=2∼ 2n ·(12− 1√2pin+O(n−32)), (3.2.21)with a denominator of[zn]D(z;2)∼ 2n√2pin·(1+O(n−1)). (3.2.22)This results in a mean number of contacts give by〈ma〉n =√pin2+O(1). (3.2.23)40Combining all three cases, we have〈ma〉n =a+22−a +O(n−1) a < 2√pin2 +O(1) a = 2a−22(a−1) (n+O(1)) a > 2. (3.2.24)In the limit as n→ ∞, we observe that the density of contacts, given by 〈ma〉nnapproaches zero when a ≤ 2, meaning that the system stays away from the wallin a desorbed phase (Figure 3.10). When a > 2, we observe a positive densityof contacts meaning that the walk remains close to the wall in an adsorbed phase(Figure 3.11).Figure 3.10: When a is small, the model is in a desorbed phase. The walktypically drifts away from the wall and has a zero density of contactsin the limit as n tends to infinity.Figure 3.11: When a is large, the model is in a adsorbed phase. The walktypically stays close to wall and has a positive density of contacts inthe limit as n tends to infinity.413.3 Alternate SolutionAn alternate technique for finding the generating function relies on factoring theDyck path each time it returns to the wall. Starting with a Dyck path of lengthn, we can prepend an up step (1,1) and appending a down step (1,−1) to obtaina new Dyck path of length n+ 2 (Figure 3.12). This new Dyck path has the spe-cial property that it does not have any wall visits other than the initial and finalvertices. These are known as the primitive Dyck paths as every Dyck path can beconstructed using a sequence of these primitive paths (For example, Figure 3.13).As a functional equation, we getD(z) = 11− z2D(z) , (3.3.1)which we can solve to recover Equation 3.1.14.Figure 3.12: A Dyck path (Left) can be made into a primitive path byprepending and appending steps (Right).Figure 3.13: Partitioning a Dyck path into primitive pieces.This idea can be use to recover the interacting model. We define the Boltzmannweight a for the interaction for wall visits. Since we omit the initial wall visit, eachprimitive piece has a single wall visit (Figure 3.14). Hence we get the functional42equationD(z;a) = 11−az2D(z;1) . (3.3.2)By making the observation that D(z;1) = D(z), we can substitute a = 1 into Equa-tion 3.3.2 recover the Catalan generating function Equation 3.1.14. We also re-cover D(z;a) (Equation 3.2.11) by substituting the Catalan generating function intoEquation 3.3.2 for D(z;1).Figure 3.14: Primitive paths have a single wall interaction.433.4 Properties of Free EnergyWe take a small detour at this point to explore an important property of the freeenergy. Figure 3.7 shows explicitly that the free energy is a continuous function inthe parameter a. The property that the free energy is continuous in its interactionparameters hold in a much more general setting, even if we are unable to explicitlydetermine the free energy. In fact, it can be shown that this property for a freeenergy κ holds for a range of models.Unfortunately, this discussion has to be separated into two different sectionsof the manuscript for the purpose of clarity. In this section, we will provide thedetails for the single interaction parameter case for a single walk in the half plane.The discussion continues in Section 4.3. The second portion will extend the re-sults of this section to models with multiple interaction parameters, (That is, thesingle walk model in a slit). The arguments in this section and that of Section 4.3are closely related to the results for self-avoiding walk models detailed in [71].Janse van Rensburg [47] contains an alternate formulation of the continuity resultsin this section starting with the general notion of convexity.3.4.1 Single Interaction ParameterWe start by looking at the partition function for a Dyck path in the half plane(Equation 3.2.2) and consider the exponential form of the Boltzmann weight α =εakbTto getZn(α) = ∑ϕ∈Dneαm(ϕ) = ∑p≥0dn(p)eα p, (3.4.1)where dn(p) is the number of elements ϕ such that |ϕ|= n and m(ϕ) = p (Defini-tion 2.7). To avoid any potential issues with parity, we only consider even valuesof n.Ultimately, we want to show that the free energy in a continuous function in theinteraction parameter a = eα . However, the argument is more clear when presentedin terms of α . Therefore, we first build towards Theorem 3.14, which will showthe free energy κ is a continuous function in the Boltzmann weight α . We thenapply a continuous transformation from α to a to obtain the result we want.The key observation that will drive what follows is that for integers u,v≥ 0 and44a fixed α , the model satisfies the inequalityZu(α) ·Zv(α)≤ Zu+v(α). (3.4.2)Suppose we start with two Dyck paths ϕu ∈Du and ϕv ∈Dv of length u and vrespectively. If we append ϕv to ϕu, we obtain a Dyck path of length u+ v. Thatis, the concatenation ϕuϕv = ϕ ′ is an element of Du+v (Figure 3.15). Since we donot weight the initial wall interaction, we have the total number of interactions isgiven by m(ϕ ′) = m(ϕu)+m(ϕv). Hence, we haveZu(α) ·Zv(α) =∑ϕueαm(ϕu) ·∑ϕveαm(ϕv) (3.4.3)=∑ϕu∑ϕveαm(ϕu) · eαm(ϕv) (3.4.4)=∑ϕu∑ϕveα[m(ϕu)+m(ϕv)] (3.4.5)=∑ϕ ′eαm(ϕ′). (3.4.6)Figure 3.15: Concatenation of two Dyck paths ϕu with m(ϕu)= 3 and ϕv withm(ϕv) = 4 to form a new Dyck path ϕ ′ of length u+ v with m(ϕ ′) =3+4 = 7.45The right hand side is a summation over the set of Dyck paths of length u+ vobtained via the concatenation process and is clearly a subset of all Dyck paths oflength u+ v. Hence, the inequality holds.In fact, this observation holds in a much wider setting. In addition to being truefor all the models in this manuscript, this observation is true for a range of non-directed walk models. In particular, it holds for the partially directed walk models[12], as well as a subset of the self-avoiding walk model [29]. With additionalwork, Hammersley et al. [28] have shown that this subset of self-avoiding walkshas the same free energy as the entire set of self-avoiding walks.We begin with a few general definitions and results that are not specific to thesemodels.Definition 3.3. A function f : R 7→ R is said to be subadditive if for x,y ∈ R wehavef (x+ y)≤ f (x)+ f (y). (3.4.7)Similarly, we can define a submultiplicative function as followsDefinition 3.4. A function f : R>0 7→ R>0 is said to be submultiplicative if forx,y ∈ R>0 we have:f (x · y)≤ f (x) · f (y). (3.4.8)While the notion of subadditive and submultiplicative functions is defined in awider scope, the stated notions are sufficient for our purposes.Lemma 3.5 (Fekete’s Lemma [21]). Let f (n) be a subadditive function in n ∈ N.The limit limn→∞f (n)nexists and is equal to inff (n)n.Proof. First, define L = infn∈Nf (n)n. Then for any ε > 0, there exists K ∈N such that∣∣∣∣f (K)K−L∣∣∣∣<ε2. (3.4.9)Pick M large enough such that for each 0 ≤ r < K, we have f (r)KM < ε2 . For everyn ≥ KL, we can write n = Kq+ r for some integers q,r such that 0 ≤ r < K and46q≥M. From this, we getf (n)n= f (Kq+ r)Kq+ r (3.4.10a)≤ f (Kq)Kq+ r +f (r)Kq+ r (3.4.10b)≤ q · f (K)Kq+ r +f (r)Kq+ r (3.4.10c)≤ q · f (K)Kq+ f (r)Kq(3.4.10d)= f (K)K+ f (r)Kq(3.4.10e)≤(L+ ε2)+ ε2(3.4.10f)= L+ ε. (3.4.10g)Combined with our definition of L, we have∣∣∣f (n)n −L∣∣∣≤ ε for n≥ KM and hencelimn→∞f (n)n= L. (3.4.11)Theorem 3.6 (Cauchy-Schwarz inequality). Let {ai}0≤i≤n, and {bi}0≤i≤n be twosequences of positive real numbers. Then(n∑i=0aibi)2≤n∑i=0(ai)2n∑i=0(bi)2 . (3.4.12)Proof. Consider the following functionQ(x) =n∑i=0(aix−bi)2 (3.4.13)= x2 ·n∑i=0(ai)2−2x ·n∑i=0aibi +n∑i=0(bi)2 . (3.4.14)Since Q(x) is a sum of quadratics, we have Q(x) ≥ 0 for any choice of x withequality exactly if x = biai for every i. In general, we have Q(x) > 0 implying that47we have a negative discriminant. That is4 ·(n∑i=0aibi)2−4 ·n∑i=0(ai)2 ·n∑i=0(bi)2 < 0 (3.4.15)(n∑i=0aibi)2<n∑i=0(ai)2 ·n∑i=0(bi)2 . (3.4.16)Combining with the case of equality, we have the desired result.Theorem 3.7. The free energy κ(α) exists for all α ∈ R.Proof. This is an direct application of Fekete’s Lemma (Lemma 3.5). For any fixedα ∈ R, the function f (n) =− log(Zn(α)) is subadditive by Equation 3.4.2. Hencelimn→∞−1nlog(Zn(α)) exist. This shows the existence of −κ(α), implying that κ(α)exists.Definition 3.8. A function f is said to be midpoint convex if it satisfies the inequal-ityf (x)+ f (y)2≥ f(x+ y2). (3.4.17)Lemma 3.9. For any fixed integer n ≥ 0, the function log(Zn(α)) is a midpointconvex function in α .Proof. To show that log(Zn(α)) is convex, we will show that for α,β ∈R, we havelog(Zn(α))+ log(Zn(β ))2≥ log(Zn(α+β2)). (3.4.18)48We apply Cauchy’s inequality as followsZn(α) ·Zn(β ) =(∑m≥0dn(m)eαm)·(∑q≥0dn(q)eβq)(3.4.19a)= ∑m≥0(√dn(m)eα2 m)2·∑q≥0(√dn(q)eβ2 q)2(3.4.19b)≥(∑m≥0√dn(m)eα2 m√dn(m)eβ2 m)2(3.4.19c)=(∑m≥0dn(m)eα+β2 m)2(3.4.19d)=(Zn(α+β2))2. (3.4.19e)Taking logarithm of both sides gives us the desired inequality. This shows log(Zn(α))is a midpoint convex function.Corollary 3.10. The function κ(α) is midpoint convex.Proof. For each n ∈ N, we havelog(Zn(α))+ log(Zn(β ))2≥ log(Zn(α+β2))(3.4.20a)1n· log(Zn(α))+ log(Zn(β ))2≥ 1nlog(Zn(α+β2))(3.4.20b)1n log(Zn(α))+ 1n log(Zn(β ))2≥ 1nlog(Zn(α+β2)). (3.4.20c)For each α and β , Fekete’s lemma guarantees the existence of the limit as n goesto infinity and so we haveκ(α)+κ(β )2≥ κ(α+β2). (3.4.21)Lemma 3.11. For a bounded interval I of α , the function Zn(α) is bounded.49Proof. For a fixed n ≥ 0, the partition function Zn(α) is a polynomial in eα ofdegree n2 . By the boundedness Theorem, we have that Zn(α) must also be bounded.We can refine Lemma 3.11 and provide bounds on Zn(α) without too muchwork. We start by bounding the value of Zn(α) for a fixed value of α . To get theupper bound, we note that a walk of length n can have at most n2 contacts with thewall (See Figure 3.16). Further, each walk is simply a sequence of up and downsteps, giving us at most 2n walks. This givesZn(α) = ∑m≥0dn(m)eαm (3.4.22a)≤ ∑m≥0dn(m)max{1,eα n2}(3.4.22b)= max{1,eα n2}∑m≥0dn(m) (3.4.22c)≤max{1,eα n2}·2n. (3.4.22d)Hence, we have Zn(α)≤max{1,eα n2}·2n.To obtain the reverse inclusion, note that the walk with exactly n2 contacts withthe wall (See Figure 3.16) is one of the summands of Zn(α). Since each walkcontributes a non-negative weight Zn(α), we get Zn(α)≥ eαn2 .For any α ∈ I, we can bound Zn(α)eαn2 ≤ Zn(α)≤max{1,eα n2}·2n. (3.4.23)These bounds can then be extended to the interval I.Figure 3.16: A dyck path of length 18 with maximal wall contacts.50To simplify notation, we define: αmax = max{0, α2}.Corollary 3.12. For a bounded interval of α , the free energy κ(α) is a boundedfunction.Proof. We make use of the bound obtained aboveκ(α) = limn→∞1nlog(Zn(α)) (3.4.24a)≤ limn→∞1nlog(eαmaxn ·2n) (3.4.24b)= limn→∞log(eαmax ·2) (3.4.24c)= limn→∞αmax + log(2) (3.4.24d)= αmax + log(2) . (3.4.24e)To get the reverse inclusion, we use Zn(α)≥ eαmaxn to get that κ(α)≥ αmax.The following result combines all we have shown about the free energy. Wefollow a modified version of the proof given in Donoghue [18].Theorem 3.13. If a function f : R 7→ R is midpoint convex and bounded on anopen interval (a,b), then f is continuous on (a,b).Proof. We approach via proof by contradiction. Suppose there exists a discontinu-ity at x = c. Without loss of generality, we can shift the function such that c = 0and f (c) = 0. Since c = 0 is a discontinuity, there exists a sequence {xn}n≥0 suchthat limn→∞xn = 0 and limn→∞f (xn) = m 6= 0. We first consider the case m > 0. Thesequence {2xn}n≥0 also converges to 0 and we havef (xn) = f(0+2xn2)(3.4.25a)≤ f (0)+ f (2xn)2(3.4.25b)= f (2xn)2. (3.4.25c)51Hence we havef (2xn)≥ 2 f (xn) . (3.4.26)This impliesliminf f (2xn)≥ 2m. (3.4.27)and through repeated iterations, we getliminf f(2kxn)≥ 2km, (3.4.28)which is impossible since our assumption is that f is a bounded function. In thecase where m < 0, we consider the inequalityf (0) = f(xn +(−xn)2)(3.4.29a)≤ f (xn)+ f (−xn)2(3.4.29b)2 f (0)≤ f (xn)+ f (−xn) (3.4.29c)0≤ f (xn)+ f (−xn) (3.4.29d)− f (xn)≤ f (−xn) . (3.4.29e)limx→∞− f (xn)≤ limx→∞f (−xn) (3.4.29f)−m≤ limx→∞f (−xn) (3.4.29g)0 < limx→∞f (−xn) . (3.4.29h)(3.4.29i)We can then use the same approach as per the previous case with m > 0 with thesequence {−xn}.Theorem 3.14. For any open bounded interval of α , the free energy κ(α) is amidpoint convex and continuous function.Proof. We have shown that κ is a midpoint convex and bounded functions in Corol-lary 3.10 and Corollary 3.12. These are the necessary conditions in Theorem 3.1352to show κ is continuous.At this point, we make the continuous transformation a = eα to get:Corollary 3.15. The limiting free energy κ(a) = limn→∞1nlog(Zn(a)) is a continuousfunction for any bounded interval of a.Before we turn our attention to the next model, we look at one very importantaspect of the free energy.Theorem 3.16. The limiting free energy κ(α) contains a point of non-analyticity.Proof. We will show that the behaviour of κ(α) behaves differently when α < 0and when α > 0. Hence, there must be a change in the nature of κ .• We will show that for α < 0, the function κ is constant. First, consider aDyck path of length n− 2. We can prepend an up step and append a downstep to form a Dyck path of length n with no wall contacts except the firstand final steps. This will create subset of all walks of length n. Hence wehave Zn−2(0) ≤ Zn(α). We bound Zn(α) from above by noting that α < 0,we have Zn(α)≤ Zn(0). Hence we haveZn−2(0)≤ Zn(α)≤ Zn(0) (3.4.30a)limn→∞1nlog(Zn−2(0))≤ limn→∞1nlog(Zn(α))≤ limn→∞1nlog(Zn(0)) (3.4.30b)κ(0)≤ κ(α)≤ κ(0). (3.4.30c)Hence we haveκ(α) = κ(0) = Constant. (3.4.31)In particular, κ(0) refers to the non-interacting case covered in Equation 3.1.14.Hence, for α < 0, we haveκ(α) = κ(0) = log(2). (3.4.32)• In the case where α > 0. We recall from Lemma 3.11 that e αn2 ≤ Zn(α) ≤532neαn2 . Hence we havelimn→∞1nlog(eαn2)≤ limn→∞1nlog(Zn(α))≤ limn→∞1nlog(2neαn2)(3.4.33)α2≤ κ(α)≤ α2+ log(2) (3.4.34)These two bounds define a region which the free energy function can exist. Thisguarantees a change in behaviour over the two regions and hence discontinuity inκ or one of its derivatives (Figure 3.17).Figure 3.17: Bounds for κ(α) showing that there is a change in behaviour.This point of non-analyticity is the phase transition. After the continuous trans-formation a= eα , we see that this phase transition happens in the region a≥ 1. Thisphase transition can be observed directly in in Figure 3.7 and Figure 3.8, whereD(z;a) exhibits two sources of singularities that swap roles when a = 2.54Chapter 4Directed Walk in the SlitThis chapter covers a natural extension of Chapter 3 with the inclusion of a sec-ond wall. This new barrier provides additional challenges when dealing with thefunctional equation by increasing the number of boundary terms. However, therestriction on the width allows us to make use of more solutions to the kernel. Sec-tion 4.1 begins by looking at introducing a second wall at width y = w to confinethe walk to a slit of width w in a non-interacting model. Section 4.2 extends themodel by allowing the walk to interact with both walls.This model was first studied by Brak et al. [10] using a very different construc-tion. Their construction used an argument that builds up configurations in a strip ofwith w+1 from configurations of width w. A recurrence-functional equation wasestablished based on this argument and they were able to obtain the exact solutionsfor three different classes of walks: loops,bridges, and tails. In all three classes thewalk begins at one wall but differ according to the end point. For loops, the walkends at the same wall. For bridges, the walk ends at the opposite wall. For tails,the walk is allowed to end at any height.This chapter considers the same model using a kernel method approach that isconsistent with the rest of the manuscript. We reproduce the results for the class ofloops in Brak et al. [10]. While we do not explicitly cover the remaining two caseshere, Theorem 1 of Brak et al. [10] shows that all three classes, loops bridges andtails, have the same limiting free energy.554.1 Non-Interacting ModelWe begin with the Dyck path model in Chapter 3 and introduce a second wall at afixed height y = w. This confines the Dyck path into a slit (Figure 4.1). Analogousto the previous model, we first consider the non-interacting case and introduce theinteractions by modifying the model accordingly.Figure 4.1: A Dyck path prefix confined to a slit of width w = 3.Analogous to the one wall case, let D∗ be the set of Dyck path prefixes wherethe maximum height is at most w. The generating function D(z,s) can be definedas follows.Definition 4.1. For ϕ ∈ D∗, let h(ϕ) denote the height of the final vertex in thewalk ϕ . The associated bivariate generating function is given byD(z,s) = ∑ϕ∈D∗z|ϕ|sh(ϕ). (4.1.1)In its full generality, D(z,s) is dependent on the width w. However, we willconsider w as a fixed parameter at this point and write it implicitly. Applying thecolumn by column construction from Chapter 3, we getD(z,s) = 1+ z(s+ 1s)·D(z,s)− zs· [s0]D(z,s)− zsw+1 · [sw]D(z,s). (4.1.2)The left hand side represents all possible walks in the system. The first threeterms on the right hand side is identical to those from the non-interacting half plane56case (Equation 3.1.2) and represents the set of walks in the half plane. However, wewant to impose the condition that the walk does not step above the line y = w. Weaccomplish this by subtracting the contribution due to the up step to walks endingat height w (Figure 4.2). This process is analogous to the boundary condition atthe wall y = 0 where we subtract the contributions of down step for walks endingat height 0. The contribution of these walks are given by zsw+1 · [sw]D(z,s).Figure 4.2: Forbidden steps taking the walk above the wall y = w.Following the previous models, we simplify to getK(z,s) ·D(z,s) = 1− zs·D0(z)− zsw+1 ·Dw(z). (4.1.3)We arrive at the analogous function to Equation 3.1.5. However, the right handside now involves two unknown functions D0(z) and Dw(z). The function D0(z) isof interest, meaning that we should eliminate Dw(z). One way we can accomplishthis is to use make use of symmetries in the kernel.This model has an identical kernel to that of the one wall model (Equation 3.1.4).Hence, we have the same solutions sˆ1 and sˆ2 (Equation 3.1.7a and Equation 3.1.7b)as the solutions to the kernel equation (K(z, sˆ) = 0). Recall for the previous modelthat the coefficients [zn]D(z,s) is a polynomial of degree n since each walk of lengthn can be at most height n. The situation is slightly different in this model. Sincethe walk is confined to a slit, the coefficient [zn]D(z,s) = Pn(s) is a polynomial ofdegree at most w. In fact, we can explicitly state the degree of Pn(s) as min{n,w}.57The upper bound on the degree of Pn(s) allows the use of both sˆ1 and sˆ2 as validsubstitutions into D(z,s) asznPn(sˆ1) = O(zn), (4.1.4)znPn(sˆ2) = O(1) when n≤ w, (4.1.5)znPn(sˆ2) = O(zn−w) when w < n. (4.1.6)There is a small caveat attached to the Equation 4.1.6. In cases where n and ware of different parity (n 6= w (mod 2)), the leading term in znPn(sˆ2) is of order onehigher than stated in Equation 4.1.6. The reason for this offset follows immediatelyfrom the observation that a walk of length n cannot end at height w if they havedifferent parities.In either case, bothD(z, sˆ1) =∞∑k=0Pk(sˆ1)zk, (4.1.7)D(z, sˆ2) =∞∑k=0Pk(sˆ1)zk. (4.1.8)produces convergent power series as the coefficients [zn]D(z, sˆ1) (and [zn]D(z, sˆ2))is the sum of a finite number of znPn(sˆ1) (and znPn(sˆ2)) respectively. This allowsus to create two different equations by substituting the two kernel solutions intoEquation 4.1.3.We take a small aside at this point to highlight a general relation between thekernel solutions. It can be easily verified thatsˆ1 =1sˆ2. (4.1.9)This is a result of the invariance of the kernel under the transformation s 7→ s−1.This symmetry gives a set of solutions sˆ such that K(z, sˆ) = 0. Therefore, wecan choose apply the symmetry in terms s as a free variable to generate a set offunctional equations as opposed to generating the same set of equations using theset of solutions sˆ.With this mind, we apply the transformation s 7→ s−1. to obtain the functional58equationK(z,s) ·D(z, 1s)= 1− zs ·D0(z)−zsw+1·Dw(z). (4.1.10)The symmetry of the kernel means the same set of functional equations willresult regardless of our choice of substitutions sˆ1 or sˆ2. Hence, we are no longerrequired distinguish between kernel solutions and we can safely omit the subscriptson sˆ without ambiguity. We make the substitution s = sˆ to eliminate the kernel andEquation 4.1.3 and Equation 4.1.10 reduces to0 = 1− zˆsˆ·D0(z)− zˆsˆw+1 ·Dw(z), (4.1.11)0 = 1− zˆsˆ ·D0(z)−zˆsˆw+1·Dw(z). (4.1.12)As noted previously, the same set of equations will result if we were to sub-stitute sˆ1 and sˆ2 separately into Equation 4.1.3. For these two equations, we caneliminate Dw(z) and after solving for D0(z),D0(z) =(sˆ2w+2−1)(sˆ2 +1)sˆ2w+4−1 . (4.1.13)While the function D0 is a function of z, it can be more compactly stated as afunction sˆ = s(z). For any given width w, we recover the explicit z dependence ofD0 by substituting the appropriate sˆ = s(z). In its stated form, we are able to extendthis further to find the dependence of the dominant singularity on the width w.In principle, Equation 4.1.13 exhibits two sources of singularities.1. Recall that sˆ = s(z) (Equation 3.1.7a and Equation 3.1.7b) is a function of zand contain a square root singularity at zc = 12 .2. There is an additional source of singularities from the zeros of the denomi-nator in terms of sˆ, when sˆ2w+4 = 1However, we show that the only singularities are of the second form. That is,the point z = 12 does not contribute to the set of singularities. To show this, we willshow that Equation 4.1.13 is a rational function and then show that z = 12 is not asingularity of the function.59First, we rewrite Equation 4.1.13 asD0(z) =(sˆw+1− 1sˆw+1)(sˆ+ 1sˆ)sˆw+2− 1sˆw+2. (4.1.14)By applying the relation between the two solutions of the kernel (Equation 4.1.9),we can writeD0(z) =(sˆw+11 − sˆw+12)(sˆ1 + sˆ2)sˆw+21 − sˆw+22. (4.1.15)For a fixed width, D0(z) is a symmetric rational function in the variables sˆ1 andsˆ2. Hence, D0(z) can be expressed as a rational function in the elementary sym-metric functions. We can easily verify that for the elementary symmetric functionson two variables, we havesˆ1 · sˆ2 = 1, sˆ1 + sˆ2 =1z. (4.1.16)By rewriting D0(z) as a rational function in the elementary symmetric function andapplying the substitutions into 1 and z−1, we see that D0(z) must be a rational in z(possibly after clearing denominators).Further, the a rational function can only contain pole type singularities. Hence,the function D0(z) must be infinite at the poles. In the limit z→ 12 , we see thatsˆ→ 1 andlimsˆ→1D0(z) =(sˆ2w+2−1)(sˆ2 +1)sˆ2w+4−1 =2w+2w+2 , (4.1.17)which is finite. Hence, we conclude that D0(z) is a rational function and is notsingular at z = 12 . Further, The only source of singularities comes from the zerosof the denominator of Equation 4.1.13.Equivalently, we can also show D0(z) is a rational function by translating thefunction into a problem of counting paths using a finite transfer matrix (For exam-ple, see Section 2.2 or Chapter V of [23]).From Equation 4.1.13, we know sˆ must be a (2w+4)-th root of unity. To avoidany cancellation with the numerator, we know that sˆ cannot be ±1 nor a (2w+2)-th root of unity. With this information, we obtain a superset of the zeros in the60denominator assˆ ∈{exp[2pii j2w+4] ∣∣∣∣ j = 1,2, . . . ,2w+4}. (4.1.18)We recover zc by solving for z in the kernel equation K(z, sˆ) = 0. Explicitly,we havez(sˆ) = 1sˆ+ 1sˆ. (4.1.19)Substituting our known values of sˆ, we recoverz(sˆ) ∈12cos(pi jw+2)1≤ j≤2w+4. (4.1.20)The singularity of smallest modulus is given byzc(w) =12cos( piw+2) . (4.1.21)At this point, we express the width w as an explicit parameter because this resultshows the dominant singularity varies as a function of width. We extract the freeenergy given byκ(w) = log[2cos(piw+2)]. (4.1.22)The width dependent dominant singularity and free energy implies the be-haviour of the system depends on the width. This naturally leads to the question ofhow the behaviour of the system changes as a function of width.One question of interest is to consider the behaviour of the system as the widthgoes to infinity.Definition 4.2. Given a model in a slit of width w with partition function Zn(w)and free energy κ(w), the infinite slit limit is defined asκinf−slit(u) = limw→∞κ(w) = limw→∞limn→∞1nlog(Zn(w)) . (4.1.23)In the infinite slit limit, we take the size of the walk to infinity first, followed61by the width. This ordering allows the system to “experience” the presence of bothwalls. In the next model where wall interactions included, the system experiencesinteractions with both sides of the slit in this limit.We could switch the order of the limits. This would mean taking the widthto infinity first, followed by the size of the walk. In this case, the walk will notinteract with the second wall at infinite width. This limiting case is known as a halfplane limit.In this model, we take the limit w→ ∞ to obtain the infinite slit limitκinf−slit = limw→∞log[2cos(piw+2)]= log(2) . (4.1.24)In the infinite slit limit, we recover the free energy obtained in the originalDyck path model. Coincidentally, this also happens to the half plane limit for thismodel.Another statistic that is of interest is how the free energy changes a function ofwidth. In statistical mechanics, this quantity is known as the effective force on thewalls exerted by the polymer. It is a measure of the “push” or “pull” on the wallsdue to the entropic expansion of the polymer.Definition 4.3. Given a model in a slit of width w with free energy κ(w), theeffective force between the walls is given by FF (w) = ∂κ(w)∂w . (4.1.25)In cases where the dominant singularity zc(w) of the grand canonical partitionfunction is known, this is equivalent toF (w) = 1zc(w)−∂ zc(w)∂w . (4.1.26)For this model, we haveF (w) = ∂κ∂w =pi(w+2)2 tan(piw+2). (4.1.27)In particular, the force is positive for all widths w. This implies the walls62experience a repulsive force at all finite widths.4.2 Interacting ModelThis next model covered is the model of a single polymer in a slit that is allowedto interact with both walls. This Dyck path model is the natural combination ofthe two previous extensions (Section 3.2 and Section 4.1). It was studied using amatrix construction by DiMarzio and Rubin [17] and via a recurrence-functionalequation construction by Brak et al. [10]. We provide an analysis of this modelvia the kernel method as a illustrative example to the key ideas for later modelsas well as developing a consistent method for analysing directed polymer models.The results we obtained via the kernel method are in agreement with the resultsobtained by Brak et al. [10]Starting with the set of Dyck path prefixes D∗ with height at most w, we addthe energies −εa and −εb for each visit of the walk to the bottom and top wallsrespectively. This gives the walk ϕ a Boltzmann weight a = exp[εakbT]for everytime the walk touches the lower wall y = 0, and with b = exp[εbkbT]for every timethe walk touches the upper wall y = w (Figure 4.3). We omit the weight of theinitial contact at (0,0).Figure 4.3: A Dyck path prefix ϕ of length |ϕ|= 6 confined to a slit of widthw = 3 with interaction parameters a,b.The associated generating function D(z,s;a,b) can be defined analogous by63extending Definition 3.2 appropriately. Namely,D(z,s;a,b) = ∑ϕ∈D∗ama(ϕ)bmb(ϕ)sh(ϕ)z|ϕ|, (4.2.1)where ma(ϕ) and mb(ϕ) is the number of times ϕ visits the bottom and top wallsrespectively.We use the column by column approach to create a functional equation satisfiedby D(z,s;a,b),D(z,s;a,b) = 1+ z(s+ 1s)·D(z,s;a,b)− zs· [s0]D(z,s;a,b)− zsw+1 · [sw]D(z,s;a,b)+ z(a−1) · [s1]D(z,s;a,b)+ zsw(b−1) · [sw−1]D(z,s;a,b). (4.2.2)As with Section 4.1, the function D(z,s;a,b) depends on the width w. However,we will treat w as a constant at this point and consider its dependence later on.Recall that Equation 4.1.2 accounts for the possible walks without interactions.We now incorporate the interaction parameters as per the single wall case (SeeEquation 3.2.3). That is, we replace the original non-weighted versions of thewalks we want to weight with interaction parameters a and b for the bottom andtop walks respectively. This givesz(a−1) · [s1]D(z,s;a,b) and zsw(b−1) · [sw−1]D(z,s;a,b). (4.2.3)We rearrange Equation 4.2.2 as per the previous models to get the kernelK(z,s) ·D(z,s) = 1− zs·D0(z)− zsw+1 ·Dw(z)+ z(a−1) ·D1(z)+ zsw(b−1) ·Dw−1(z). (4.2.4)This function contains four unknown functions on the right hand side. We followthe method of reducing the unknowns in Equation 3.2.6. By extracting the coeffi-cients [s0] and [sw] of Equation 4.2.4, we get a relation between D1 and D0, as well64as a relation between Dw−1 and Dw.D0(z) = 1+ zaD1(z), (4.2.5)Dw(z) = zbDw−1(z). (4.2.6)These equations can be understood as follows. Every walk ending at the bot-tom wall is either the trivial walk or obtained by appending a step to a walk endingat height 1 with appropriate weights. Similarly, every walk ending at the top wall(height w) is obtained by appending a step to a walk of height w−1 with appropri-ate weights. By eliminating D1(z) and Dw−1(z), Equation 4.2.4 simplifies toK(z,s) ·D(z,s) = 1a− sa− s− zasa·D0(z)−sw(szb−b+1)b·Dw(z). (4.2.7)Since Equation 4.2.7 only contains D0(z) and Dw(z), it is analogous to Equa-tion 4.1.3 in the non-interacting case and we can apply the kernel symmetry s 7→ 1sto eliminate Dw(z) by choosing sˆ = s(z) such that K(z, sˆ) = 0. After rearranging,we getD0(z;a,b) =Pw(z;0,b)Pw(z;a,b), (4.2.8)where Pw(z;a,b) is a polynomial in sˆ given byPw(z;a,b) = (sˆ2− (b−1))(sˆ2− (a−1))sˆ2w− (sˆ2(b−1)−1)(sˆ2(a−1)−1).(4.2.9)Similar to solution of the non-interacting model (Equation 4.1.13), the functionD0(z;a,b) is a symmetric function in sˆ1 and sˆ2. Hence, it must be a rational functionin z. Since it is more compactly stated as a rational function in sˆ, we will derive theresults in sˆ and then map it back to z using the kernel using Equation 4.1.19Equation 4.2.8 shows that a superset of the singularities in D0(z;a,b) is the setof zeros of Pw(z;a,b). Rearranging Equation 4.2.9 to find zeros, we getsˆ2w = (sˆ2(b−1)−1)(sˆ2(a−1)−1)(sˆ2− (b−1))(sˆ2− (a−1)) . (4.2.10)This reduces the task of finding the singularities of D0(z;a,b) to finding solutions65to Equation 4.2.10.Theorem 2.14 provides one immediate consequence of this equation. By choos-ing h(s)= (s2−(a−1))(s2−(b−1)), n= 4, and d = 2w+4, Theorem 2.14 applieswhenever the zeros of h(s) lie inside the unit disc. Solving for h(s), we gets =±√a−1,±√b−1, (4.2.11)which means the zeros will be inside the unit disc when a,b ≤ 2. Hence the solu-tions to Equation 4.2.10 will be on the unit circle when both a,b < 2.In the subsequent sections, we present the results obtained using Equation 4.2.10in different regions of the a−b interaction space. Exact results were obtained fora,b ∈ {1,2} as well as the curve ab− a− ab = 0. Asymptotic results are pre-sented in the remaining cases where we were unable to find closed form results. Ineach section, we present a brief description of how the solutions are obtained forsˆ. We then state the equivalent result in z (using Equation 4.1.19), the associatedfree energy (Definition 2.28) and exerted forces on the wall (Definition 4.3). Forcomparison, the results obtained using the kernel method mimic those obtained inBrak et al. [10] where the variable q they used can be mapped to the variable usedhere via q = sˆ2.4.2.1 Case (I): a = 1,b = 1This case was treated in Section 4.1. After substitution, Equation 4.2.10 simplifiesto sˆ2w+4 = 1 implying that sˆ is a (2w+ 4)-th root of unity. This is in agreementwith the results obtained in Equation 4.1.18. The free energy is given byκ(w;1,1) = log[2cos(piw+2)](4.2.12)= log(2)− pi22w2+ 2pi2w3− pi4 +72pi212w4+O(w−5), (4.2.13)66with the force on the wallsF (w;1,1) = piw+2 · tan(piw+2)(4.2.14)= pi2w3− 6pi2w−4+O(w−5). (4.2.15)4.2.2 Case (II): a = 2,b = 2In this case, Equation 4.2.8 simplifies toD0(z;a,b) =(sˆ2w +1)(1+ s2)(s2−1)(s2w−1) (4.2.16)implying that sˆ is a (2w)-th root of unity. In particular, sˆ = 1 is zero of the denom-inator that does not simplify with the numerator. Hence it gives a true singularity.By mapping this back to zˆ via the kernel, we get the constantzc(w,2,2) =12. (4.2.17)This give a free energy κ(w;2,2) = log(2) that is independent of width. Further,this results in zero force exerted on the walls.4.2.3 Case (III): a = 2,b = 1In this case, Equation 4.2.10 simplifies to sˆ2w+2 =−1 which means sˆ is a subset ofprimitive (4w+4)-th roots of unity. Mapping this back to zˆ, we get the dominantsingularity atzc(w;2,1) =12cos(pi2(w+1)) . (4.2.18)The free energy is given byκ(w;2,1) = log[2cos(pi2(w+1))](4.2.19)= log(2)− pi28w2+ pi24w3− pi4 +72pi2192w4+O(w−5). (4.2.20)67with the force on the wallsF (w;2,1) = pi2(w+1)2 · tan(pi2(w+1))(4.2.21)= pi24w3− 3pi24w−4+O(w−5). (4.2.22)In the case where a = 1, b = 2, Equation 4.2.10 simplifies identically and gives theresults: zc(w;1,2)= zc(w;2,1), κ(w;1,2)= κ(w;2,1), andF (w;1,2)=F (w;2,1).4.2.4 Case (IV): a,b < 2Theorem 2.14 tells us that the solutions of Equation 4.2.10 are on the unit circle.In particular, we anticipate solutions to be perturbation of (2w)-th roots of unitybased on the results for Case (I),Case (II), and Case (III). We looked at solutionsof the formsˆ = exp[piiw(c0 +c1w+ c2w2+ · · ·)]. (4.2.23)We substituted this into Equation 4.2.10 and solve for the coefficients givingsˆ= exppiiw− 2(ab−a−b)(a−2)(b−2)1− 4pi2(ab−a−b)(ba2 +b2a−10ab+8a+8b−8)3(a−2)3(b−2)3(w− 2(ab−a−b)(b−2)(a−2))3+ O((w− 2(ab−a−b)(b−2)(a−2))−5))]. (4.2.24)After mapping back to z we getzc(w;a,b) =12+ pi24w2+ pi2(ab−a−b)(a−2)(b−2)w3+(5pi448+ 3pi2(ab−a−b)2(a−2)2(b−2)2)1w4+O(w−5). (4.2.25)The free energy is given byκ(w;a,b) = log(2)− pi22w2+ 2pi2(ab−a−b)(a−2)(b−2)w3 +O(w−4), (4.2.26)68with the force on the wallsF (w;a,b) = pi2w3+ 6pi2(ab−a−b)(a−2)(b−2)w−4 +O(w−5). (4.2.27)4.2.5 Case (V): a < 2,b = 2After substitution, Equation 4.2.10 simplifies tosˆ2w = sˆ2(a−1)−1sˆ2− (a−1) . (4.2.28)As per Case (IV), we expect the solutions of sˆ to be on the unit circle. Hence wesearch for solutions of the form in Equation 4.2.23, which results insˆ = exp[pii2(w− aa−2)(1− a(a−1)pi2(a−2)3(w− aa−2)3 +O((w− aa−2)−5))].(4.2.29)After mapping back to z, we getzc(w;a,b) =12+ pi216w2+ pi2a8(a−2)w3+(5pi4768+ 3pi2a216(a−2)2)1w4+O(w−5). (4.2.30)The free energy is given byκ(w;a,2) = log(2)− pi28w2− pi2a4(a−2)w3 +O(w−4), (4.2.31)with the force on the wallsF (w;a,2) = pi24w3+ 3pi2a4(a−2)w−4 +O(w−5). (4.2.32)Analogous results hold in the case when a = 2 and b < 2 with the roles of theparameters reversed.694.2.6 Case (VI): a > 2 and a > bIn this case, the solutions to Equation 4.2.10 is no longer restricted to the unitcircle. In particular, Equation 4.2.10 is satisfied in the large w limit by sˆ = 1√a−1and sˆ = 1√b−1 . We present the asymptotic expansion in the case where a > b. Inthe reverse case of b > a, we reverse the role of the parameters in the solution. Weexpand about the point sˆ = 1√a and found that the next term is exponential in w. Westart withsˆ = 1√a−1(1+ c0(a−1)w + · · ·). (4.2.33)We substituted this form into the left and right hand sides of Equation 4.2.10 andsolved for the coefficients to getsˆ = 1√a−1(1− a(a−2)(ab−a−b)2(a−1)(a−b)1(a−1)w +O((a−1)−2w)). (4.2.34)In terms of z, we getzc(w;a,b) =√a−1a− (a−2)2(ab−a−b)2a(a−b)√a−11(a−1)w +O((a−1)−2w). (4.2.35)The free energy is given byκ(w;a,b) = log(a√a−1)+ (ab−a−b)(a−2)2(a−1)(a−b)1(a−1)w +O((a−1)−2w),(4.2.36)with the force on the wallsF (w;a,b) =−(ab−a−b)(a−2)2 log(a−1)2(a−1)(a−b)1(a−1)w +O((a−1)2w).(4.2.37)4.2.7 Case (VII): a > 2 and a = bThe special case where a = b with a > 2 results in some simplification in Equa-tion 4.2.10. We getsˆw =− sˆ2(a−1)−1sˆ2− (a−1) . (4.2.38)70Following a similar process as the previous case, we getsˆ = 1√a−1(1− a(a−2)2(a−1)(1√a−1)w2+O((1√a−1)w)), (4.2.39)which is equivalent tozc(w;a,b) =√a−1a− (a−2)22a√a−1(1√a−1)w2+O((1√a−1)w). (4.2.40)The free energy is given byκ(w;a,b) = log(a√a−1)+ (a−2)22(a−1)1(a−1)w2+O((a−1)−w), (4.2.41)with the force on the wallsF (w;a,b) =−(a−2)2 log(a−1)4(a−1)1(a−1)w2+O((a−1)w) . (4.2.42)4.2.8 Case (VIII): ab−a−b = 0The factor ab− a− b appears in many of the above asymptotic expansions. Thissuggests interesting behaviour along this particular curve. We considered the casewhere b = aa−1 with a > b. In the case where b > a, the roles of the parameters arereversed. Substituting this into Equation 4.2.9 givesPw(zˆ;a,b) = (s2− (a−1))(s2(a−1)−1)(s2w−1). (4.2.43)The zeros of this function is exactly when sˆ = 1√a−1 , which results in the widthindependent solution ofzc(w;a,b) =√a−1a. (4.2.44)This gives a constant free energy of κ = log(a√a−1)with zero force exerted on thewalls.714.2.9 DiscussionA detailed analysis of the results is presented at the start of Chapter 6 that will serveas an introduction and motivation for the two walk model. Therefore, we present abrief discussion at this point to serve as a summary of the results we obtain in thissection.While we were unable to determine the closed form solution for finite widths,it is possible to calculate the infinite width limits for free energy of the modelEquation 4.1.24 by taking the limit w→ ∞.κinf−slit(a,b) = limw→∞κ(w;a,b) =log(2) a,b≤ 2log(a√a−1)a > 2 and a≥ blog(b√b−1)b > 2 and b > a(4.2.45)This shows the existence of three phases: The walk is desorbed from bothwalls, adsorbed to the bottom wall, and adsorbed to the top wall (See Figure 4.4).Further, it highlights the importance of the parameter values a,b = {1,2} as wellas the line a = b.In a similar light, we can obtain a force diagram for the a−b parameter space(See Figure 4.5). In this model, we observe both attractive and repulsive regions.The repulsive region can be further subdivided into a long range repulsion regionwhere the force decays according to a power law and a short range repulsion regionwhere the force decays exponentially. The curve ab−a−b = 0 is known as a zeroforce curve as the walks exert no force on the walls and separate the parameterspace into attractive and repulsive regions. The line a = b (dotted line), whileexerts an attractive force, experiences a different exponential decay in the width.72Figure 4.4: Phase diagram of the infinite strip showing the three observedphases.Figure 4.5: Diagram of effective forces experienced by the walls of the slitmodel.734.3 Properties of Free Energy (Continued)This section is the continuation of the discussion presented in Section 3.4. Recallfor the Dyck path model, the free energy is a continuous function in the interactionparameters. In this section, we extend the continuity of the free energy to modelswith multiple interaction parameters.4.3.1 Multiple Interaction ParametersThe single walk in a slit model is the first example of a model with multiple in-teraction parameters. In this two parameter model, we have a different interactionparameters for each of the two walls. We take the analogous approach in this sec-tion as per the single interaction parameter case in Section 3.4. We expect theresults obtained for this bivariate model to be analogous to that for the single inter-action case. We provide the details for the bivariate case as the arguments used canbe extended, almost verbatim, to the multivariate cases that we will be interestedin.We begin by considering the partition function for the slit case, which we obtainusing Equation 4.2.1.Zn(α,β ) = ∑ϕ∈D∗neαma(ϕ)eβmb(ϕ), (4.3.1)which can be rewritten using Definition 2.9 to giveZn(α,β ) = ∑p,q≥0dn,p,qeα peβq, (4.3.2)where dn,p,q is the number of elements with |ϕ|= n, ma(ϕ) = p and mb(ϕ) = q.This partition function satisfies the key observation as before with Zn · Zm ≤Zn+m. Hence, Fekete’s lemma (Lemma 3.5) applies and hence the free energyκ(w;α,β ) exists. To show that it is continuous in the parameters α and β , weproceed using a similar argument as the single parameter case. To be completelyrigourous, we have to include w as a parameter of the partition function and the lim-iting free energy. However, the width remains fixed in the following calculationsand hence we simplify notation by implicitly noting the w dependence temporarily.74Theorem 4.4. The partition function Zn(α,β ) is a midpoint convex function is itsparameters.Proof. We follow a similar argument as per Theorem 3.14. Consider the productZn(α,β ) ·Zn(γ,δ ) = ∑p≥0∑q≥0dn(p,q)eα peβq ·∑r≥0∑s≥0dn(r,s)eγreδ s= ∑p≥0(√∑q≥0dn(p,q)eβq)2eα p ·∑r≥0(√∑s≥0dn(r,s)eδ s)2eγr≥(∑p≥0[√∑q≥0dn(p,q)eβq ·√∑s≥0dn(p,s)eδ s]eα+γ2 p)2=(∑p≥0√∑q≥0dn(p,q)eβq ∑s≥0dn(p,s)eδ seα+γ2 p)2≥(∑p≥0∑q≥0√dn(p,q) ·√dn(p,q)eβ+δ2 qeα+γ2 p)2=(∑p≥0∑q≥0dn(p,q)eβ+δ2 qeα+γ2 p)2=[Zn(α+ γ2, β +δ2)]2.The two inequalities are obtained using Cauchy’s inequality. First on the outersummation and then on the inner summation. Taking logarithm of both sides yieldslog(Zn(α,β )+ log(Zn(γ,δ ))2≥ log(Zn(α+ γ2, β +δ2)). (4.3.3)We define αmax = max{0,α} and βmax = max{0,β} to show Zn(α,β ) is abounded function by following a similar argument as Lemma 3.11.Lemma 4.5. For any bounded domain, the function Zn(α,β ) is bounded.Proof. We proceed as per Lemma 3.11 by noting that the partition function Zn(α,β )is a polynomial in eα and eβ . By the boundedness Theorem, eα and eβ must be75bounded. In turn, the same theorem is applied again to show the partition function,Zn(α,β ), must also be bounded.As with Lemma 3.11, we can refine the result by finding bounds on the partitionfunction without too much work. We first note that a walk of length n can have atmost n contacts with the top and bottom walk. In principle, we can obtain a betterbound than n, but this will be sufficient for our purposes.Zn(α,β ) = ∑p,q≥0dn(p,q)eα peβq≤ ∑p,q≥0dn(p,q)eαmaxneβmaxn= eαmaxneβmaxn ∑p,q≥0dn(p,q)≤ eαmaxneβmaxn2n.To obtain the lower bound, we look at two particular walks (See Figure 4.6), thewalk with maximum number of contacts with the bottom wall(n2)and the walk withmaximum contacts with the top wall(n−w2). This means Zn(α,β )≥ eαn2 + eβ n−w2 .Figure 4.6: Walks of length 18 in a slit of width 6 with maximal wall contactswith top wall and bottom wall.As per the single parameter case, we want to show that a bounded, midpointconvex function has to be continuous. In higher dimensions, we need to extend tonotion of an interval to a convex open disc.Theorem 4.6. If a function f : R2 7→R is midpoint convex in each component andbounded on a convex open disc D, then f is continuous on D.Proof. Without loss of generality, we consider a discontinuity at z = (0,0) withf (z)= 0. There must be a sequence zn =(xn,yn) such that limn→∞zn = 0 and limn→∞f (zn)=76m 6= 0. The proof then follows along a similar argument as per the one dimensionalcase in Theorem 3.13.Combining the three previous results, we obtain the continuity condition weneed for the two interaction case.Theorem 4.7. For all α,β ∈ R and a fixed width w, the free energy κ(α,β ) is aconvex and continuous function of α,β . Further, the functions free energy κ(a,b)and the partition function Zn(a,b) are both continuous functions in their parame-ters a,b.In the following chapters, we will be analysing models with a greater numberof interaction parameters. So we have to extend the previous result accordingly.Provided that the key observation (Equation 3.4.2) hold in the models we study,the same line of argument as Theorem 4.6 will follow. As a result, we can showthe following for models involving a higher number of interaction parametersLemma 4.8. If a function f : RN 7→ R is midpoint convex in each component andbounded on a convex open set D, then f is continuous on D.Unfortunately, the equivalent of Theorem 3.16 does not exist. By viewing adirected walk in a slit as a discrete time evolution one-dimensional system, it hasbeen shown that the system does not exhibit any phase transitions in the parameterspace at any finite width [67].77Chapter 5Two Directed Walks Above a WallThis chapter, as well as Chapter 6, is joint work with Aleks Owczarek and An-drew Rechnitzer. We present these chapters in a manner most consistent with thepublished manuscripts [55, 74]. While both chapters are self-contained, we willinclude additional details and references to previous sections where appropriate.The adsorption of polymers on a sticky wall, and confined between two walls,has been the subject of continued interest [1, 10, 16, 46–48, 52, 53, 59, 62, 68, 70].This has been in part due to the advent of experimental techniques able to micro-manipulate single polymers [3, 63, 64] and the connection to modelling DNA de-naturation [20, 40–45, 49].Consider a polymer in a dilute solution of good solvent, so that it is in a swollenstate [15]. If such a polymer is then attached to a wall at one end the rest of thepolymer drifts away due to entropic repulsion. On the other hand, if the wall hasan attractive contact potential, so that it becomes “sticky” to the monomers, thepolymer can be made to stay close to the wall by a sufficiently strong potential,or for low enough temperatures. The phase transition between these two statesis the adsorption transition. The high temperature state is desorbed while the lowtemperature state is adsorbed. This pure adsorption transition has been well studied[16, 30, 47, 59, 69] exactly and numerically, and has been demonstrated to besecond-order.There has been recent interest [1] in ring polymers, modelled by self-avoidingpolygons, being adsorbed onto the walls of a two-dimensional slit. In that work,78models in which both sides of the polygon could interact with each of the wallswere considered. This provides us with one of the motivations for the model here,where we consider two directed walks that begin and end together, so forming apolygon. We consider directed walks because they often admit exact solutions,while the more realistic self-avoiding walks do not. Moreover, we consider sucha pair of walks interacting with a sticky wall, allowing different interactions whenone or both walks are near the wall. To allow for a simple realisation of the modelwe consider so-called friendly directed walks (rather than the ubiquitous viciouswalks) where the two walks may share edges of the lattice. However, we do notallow the walks to cross and so there is always an upper walk/polymer and a lowerwalk/polymer.Other physical motivations for two-walk models have appeared in the litera-ture. In particular, one may model DNA-denaturation in this way — for examplethe Poland-Scheraga models [57, 61]. It would be interesting to see if the tech-niques described below could be used to find exact solutions of the DNA unzippingtransition in the presence of an adsorbing wall, such as that discussed in [33]. Todo so we would add a contact interaction between the two walks, rather than thedouble-visit interaction discussed here [66].As we investigated the model another motivation for its interest became appar-ent; the full two parameter model is not amenable to one of the standard methodsof solving multiple walk models. The Lindstro¨m-Gessel-Viennot lemma [26, 38](which was also considered earlier in a probabilistic context by Karlin and McGre-gor [34]) decomposes the solution of models of multiple vicious walks into com-binations of single walk problems. The lemma implies that the generating functionof a multiple walk model would be governed by a D-finite (Differentiably-Finite)function, but the solution of our model is not D-finite. Despite this, we are stillable to solve the model.We have solved our model in two ways. Firstly, we use the obstinate kernelmethod (see [8] for an overview of the technique) to give a formal solution ofa functional equation as a constant term formula. This constant term can thenbe evaluated explicitly. Secondly, we also use a “primitive piece” decomposi-tion similar to Section 3.3 that allows us to give an explicit solution in terms ofhypergeometric-type sums.79Our solution allows us to fully analyse the model and we find a rich phasediagram. In particular, we find three phases that meet at a special point. Thereare three phase boundaries; two are second-order and one is first-order. Intrigu-ingly, we find that arguably the most physically realisable one-parameter case ofour model would have two phase transitions on lowering the temperature.In the next section (Section 5.1), we formally define our model. In Section 5.2,we formulate a functional equation obeyed by an extended generating function andprovide a “constant term” solution in Section 5.3 via the obstinate kernel method.We provide an alternate explicit solution in Section 5.4 that illustrates an under-lying structure in the solution which arises combinatorially. In Section 5.5, weanalyse the phase structure and phase transitions of the model. In the final section(Section 5.6) we discuss the functional nature of the solution and summarise ourresults by recasting them in terms of some physical parameters of a family of singleparameter models.5.1 ModelWe consider a pair of directed walks above a wall on the upper half-plane of thesquare lattice, taking steps (1,1) or (1,−1). These walks may touch but not cross;such walks are sometimes called friendly walks. Further, we consider those pairsof walks that begin at the point (0,0) and have equal length. Let ϕ be a pair ofsuch walks and the set of all such walks be Ω. We define |ϕ| to be the length of thewalks.To these configurations we add an energy −εa for visits of the bottom walkonly (single visits) to the wall, and an energy −εd when both walks visit a site onthe wall simultaneously (double visits), excluding the first vertex of the walks. Thenumber of single visits to the wall will be denoted ma(ϕ), while the number ofdouble visits will be denoted md(ϕ).Later in the chapter we will specialise to those configurations, ϕ̂ in which bothwalks start and end on the wall. Since every such configuration has at least onedouble visit (the final vertex), we have md(ϕ̂) ≥ 1. The partition function for ourmodel isZn(a,d) = ∑ϕ̂3|ϕ̂|=ne(ma(ϕ̂)·εa+md(ϕ̂)·εd)/kBT (5.1.1)80where T is the temperature and kB the Boltzmann constant, and associated Boltz-mann weights are denoted a = eεa/kBT and d = eεd/kBT . The thermodynamic re-duced free energy of our model is given in the usual fashion from Definition 2.28κ(a,d) = limn→∞1nlog(Zn(a,d)) . (5.1.2)A configuration of length 10 in our model with single and double visits markedappears in Figure 5.1.Figure 5.1: Two directed walks of length 10 of our model that begin and endon the surface. There are two single and two double visits marked. Theleft-most (start) vertex of the two walks on the wall is not counted as adouble visit.To find the free energy we will instead solve for the generating functionG(a,d;z) =∞∑n=0Zn(a,d)zn. (5.1.3)The radius of convergence of the generating function zc(a,d) is directly related tothe free energy viaκ(a,d) = log(zc(a,d)−1). (5.1.4)815.2 Functional EquationsTo find G, we consider walks ϕ in the larger set, where each walk can end at anypossible height. Let us first consider a = d = 1. In this case we construct theexpanded generating functionF(r,s;z)≡ F(r,s) = ∑ϕ∈Ωz|ϕ|rbϕcsdϕe/2, (5.2.1)where z is conjugate to the length |ϕ| of the walk, r is conjugate to the distance bϕcof the bottom walk from the wall and s is conjugate to half the distance dϕe be-tween the final vertices of the two walks. Since the distance between the endpointsof the walks changes by 0 or ±2 with each step, and the endpoints start together,it is always an even number. Further, we let [r jsk]F(r,s) denote the coefficient ofr jsk in the generating function F(r,s). We use [r j]F(r,s) to denote the coefficientof r j in F(r,s) which is a function of s and similarly [sk]F(r,s) gives a function ofr.Let us now return to general a and d. All pairs of walks can then be builtusing the column-by-column construction we used for the single walk model (SeeSection 3.2). Translating this into its action on the generating function gives thefollowing functional equationF(r,s) =1+ z(r+ 1r+ sr+ rs)·F(r,s)− z(1r+ sr)· [r0]F(r,s)− zrs· [s0]F(r,s)+ z(a−1)(1+ s) · [r1]F(r,s)+ z(d−a) · [r1s0]F(r,s). (5.2.2)This functional equation is analogous to Equation 3.2.3 for the single walkmodel. We explain each of the terms in this equation.• The trivial pair of walks of length 0 gives the initial 1 in the functional equa-tion.• Every pair of walks may be extended by appending directed steps to theirendpoints in four different ways (see Figure 5.2).82Figure 5.2: Adding steps to the walks when the walks are away from the wall.There are four possibilities.Top walk Bottom walk Generating Function(1,1) (1,1) r ·F(r,s)(1,1) (1,−1) sr ·F(r,s)(1,−1) (1,1) rs ·F(r,s)(1,−1) (1,−1) 1r ·F(r,s)Figure 5.3: The first boundary term in the functional equation removes thecontribution from the walks that are produced by appending a SE stepto the bottom walk when its endpoint is on the wall.• Appending steps in this way may result in the bottom walk stepping below83the wall (Figure 5.3). Thus, when the bottom walk is at the wall, we cannotappend any steps that will decrease the height of the bottom walk. Theseforbidden configurations are counted byz(1r+ sr)· [r0]F(r,s) = z(1r+ sr)·F(0,s). (5.2.3)Figure 5.4: The second boundary term in the functional equation removes thecontribution of walks that cross. Such configurations are produced whenone appends steps to walks that end at the same vertex as shown.• Similarly, at no time can the top walk pass below the bottom walk (Fig-ure 5.4). Thus, if the two walks are touching, we forbid the distance betweenthem to decrease. These configurations are counted byzrs· [s0]F(r,s) = zrs·F(r,0). (5.2.4)• This accounts for the possible pairs of walks without the interaction param-eters. We can now incorporate the interaction parameters. In order to dothis, we have to add in all walks we want to mark with a and subtract thenon-weighted version of those exact same walks from the model (Figure 5.5— left). In order for the bottom walk to touch the wall, it must be at height 1initially and then step down (with no restriction on the top walk). Hence we84get the termz(a−1)(1+ s) · [r1]F(r,s). (5.2.5)Figure 5.5: Configurations that lead to single (left) and double (right) inter-action terms.• A similar method can be used to incorporate d into the model (Figure 5.5 —right). One step before both walks touch the wall they will both be at height1. All such walks have already been accounted for when incorporating a intothe model and so must be replaced. This results inz(d−a) · [r1s0]F(r,s). (5.2.6)The functions [r1]F(r,s) and [r1s0]F(r,s) can be simplified in terms of F(0,0) andF(0,s) with the same arguments used to obtain Equation 3.2.6. By extracting thecoefficient of r0s0 in the functional equation, we obtainF(0,0) = 1+ zd · [r1s0]F(r,s). (5.2.7)At a combinatorial level, this states that a pair of walks that end at the wall is eithera trivial configuration or obtained by appending a pair of SE steps to the end of apair of walks that end at height 1. Similarly, we can extract the coefficient of r0 in85the functional equation to obtainF(0,s) = 1+ za(1+ s) · [r1]F(r,s)+ z(d−a) · [r1s0]F(r,s). (5.2.8)This has a similar combinatorial interpretation to the previous case. These equa-tions can then be combined to simplify the functional equation(1− z[r+ 1r+ sr+ rs])·F(r,s) =1d+(1− 1a− zsr− zr)·F(0,s)− zrs·F(r,0)+(1a− 1d)·F(0,0). (5.2.9)We will use the above form of the equation in what follows. The polynomial coeffi-cient on the left hand side is called the kernel K(r,s;z)≡K(r,s), and its symmetriesplay a key role in the solution:K(r,s) =[1− z(r+ 1r+ sr+ rs)]. (5.2.10)5.3 Solution of the Functional EquationsIn what follows we use the obstinate kernel method. The discussion below is self-contained, but we refer the reader to Chapter 3 and Chapter 4 for examples of itsapplication to walk models, as well as the paper of Bousquet-Me´lou and Mishna[8] for a general description of this technique.5.3.1 Solution of the Functional Equations When a = 1When a = 1, the functional Equation 5.2.9 simplifies toK(r,s) · rsdF(r,s) = rs− zsd (1+ s) ·F(0,s)− zr2dF(r,0)+ rs(d−1)F(0,0).(5.3.1)We use the kernel method which exploits the symmetries of the kernel to re-move boundary terms (ie the functions F(r,0) and F(0,s)) in the above equation.A detailed treatment of the kernel method for a single walk model can be found in86Section 3.1.The kernel is symmetric under the following two transformations:(r,s) 7→(r, r2s), (r,s) 7→( sr,s). (5.3.2)These transformations generate a family of 8 symmetries (sometimes referred to asthe ‘group of the walk’ — see [8])(r,s),(r, r2s),( sr, sr2),(rs, 1s),(1r, 1s),(1r, sr2),(rs, r2s), and( sr,s).(5.3.3)We make use of 4 of these transformations — those which only involve positivepowers of r. To be precise,K(r,s) · rsdF(r,s) = rs− zsd (1+ s) ·F(0,s)− zr2d ·F(r,0)+ rs(d−1) ·F(0,0);(5.3.4a)K(r, r2s)· dr3sF(r, r2s)= r3s− zdr2s(1+ r2s)·F(0, r2s)− zr2d ·F(r,0)+ r3s(d−1) ·F(0,0);(5.3.4b)K(rs, r2s)· dr3s2F(rs, r2s)= r3s2− zdr2s(1+ r2s)·F(0, r2s)− zr2ds2·F( rs,0)+ r3s2(d−1) ·F(0,0);(5.3.4c)K(rs, 1s)· drs2F(rs, 1s)= rs2− zds(1+ 1s)·F(0, 1s)− zdr2s2·F( rs,0)+ r(d−1)s2·F(0,0).(5.3.4d)All of these transformations were chosen so that the kernel remains unchanged, andso that the substitution only involves positive powers of r. We can then eliminatethe boundary terms by taking an alternating sum of the above equations:Eqn(5.3.4a)−Eqn(5.3.4b)+Eqn(5.3.4c)−Eqn(5.3.4d).(In the case where a 6= 1, a similar method holds, except that then we must multi-ply some of the equations by non-trivial coefficients to eliminate boundary terms.)87After simplification we obtainK(r,s) · (linear combination of F) =r(s−1)(s2 + s+1− r2)s2(1+(d−1)F(0,0))− zd(1+ s)sF(0,s)+ zd(1+ s)s2F(0, 1s). (5.3.5)We can now remove the left-hand side of the equation by choosing a value of rthat sets the kernel to zero — provided all the F’s on the left-hand side remainconvergent. The kernel has two roots and we choose the one which gives a positiveterm power series expansion in z with Laurent polynomial coefficients in s:rˆ(s;z)≡ rˆ =s(1−√1−4 (1+s)2z2s)2(1+ s)z = ∑n≥0Cn(1+ s)2n+1z2n+1sn, (5.3.6)where Cn = 1n+1(2nn)is a Catalan number. This is chosen so that K(rˆ,s) = 0, andso that all the various substitutions are convergent. More precisely, since rˆ = O(z),the functions F(rˆ,s),F(rˆ, rˆ2/s),F(rˆ/s, rˆ2/s) and F(rˆ/s,1/s) are all formally con-vergent power series in z with Laurent polynomial coefficients in s.We are not able to use the other root of the kernel (with respect to r) since itis O(z−1). If we were to substitute this into the functional equation, then F(r,s),F(r,r2/s), F(r/s,r2/s), and F(r/s,1/s) would not converge within the ring offormal power series. This follows since the coefficient of zn in F(r,s) has degree nin r and so substituting r 7→O(z−1) will map terms in this polynomial to all powersof z including the constant term.When we make the substitution r 7→ rˆ we can rewrite the coefficients of theright-hand side so as to not explicitly involve z — since now z=(rˆ+1/rˆ+ rˆ/s+ s/rˆ)−1.0 = rˆ(s−1)(s2 + s+1− rˆ2)s2(1+(d−1)F(0,0))− drˆs2s+ rˆ2 F(0,s)+drˆ(s+ rˆ2)s F(0, 1s).(5.3.7)Because we are primarily interested in F(0,0) — the generating function of88pairs of walks that start and end on the wall — it is convenient to rewrite theequation so that there are no powers of s or rˆ in the denominator of the coefficientsand so that the coefficients of F(0,s) and F(0,1/s) are independent of rˆ.0 = ds4F(0,s)−dsF(0, 1s)− (s−1)(s2 + s+1− rˆ2)(s+ rˆ2)(1+(d−1)F(0,0)) .(5.3.8)Consider the coefficient of s1 in the above equation, or rather by dividing theequation by s consider the constant term, that is the coefficient of s0 in the equation.This leads us to calculate F(0,0) effectively as a constant term in the variable s.For ease of calculation and display we will continue with calculating the coefficientof s1 in Equation 5.3.8.Since F(0,s) is a power series in z with polynomial coefficients in s, the termds4F(0,s) does not contain any coefficients of s1. Similarly, F(0, 1s)is a powerseries in z with polynomial coefficients in s−1, so the term dsF(0, 1s)contributesonly dF(0,0). For the remaining term, we consider the coefficient [s1](s−1)(s2 +s+1− rˆ2)(s+ rˆ2). Expanding the expression and then collecting the exponents ofrˆ gives:(s−1)rˆ4 +(1− s+ s2− s3)rˆ2 + s(1− s)(s2 + s+1). (5.3.9)We need to consider the expansion of rˆ2 and rˆ4. Lagrange inversion [23] gives:rˆ(s;z) =∞∑n=0Cn(1+ s)2n+1snz2n+1, (5.3.10a)rˆ(s;z)2 =∞∑n=0Cn+1(1+ s)2n+2snz2n+2, (5.3.10b)rˆ(s;z)4 =∞∑n=042n+4(2n+4n)(1+ s)2n+4snz2n+4, (5.3.10c)and, more generally,rˆ(s;z)k =∞∑n=0k2n+ k(2n+ kn)(1+ s)2n+ksnz2n+k. (5.3.10d)89Computing the coefficient of a particular power of s in rˆ2 or rˆ4 reduces to find-ing the coefficient of powers of s in (1+s)ns−k which are just binomial coefficients:[s1](s−1)rˆ4 =∞∑n=0−6(n−1)n(n+2)2 C2nz2n+2; (5.3.11a)[s1](1− s+ s2− s3)rˆ2 =∞∑n=06(n2 +1)(n+2)(n+3)C2nz2n+2; (5.3.11b)[s1]s(1− s)(s2 + s+1) = 1; (5.3.11c)[s1](s−1)(s2 + s+1− rˆ2)(s+ rˆ2) = 1+∞∑n=012(2n+1)(n+2)2(n+3)C2nz2n+2. (5.3.11d)Hence extracting the coefficient of s1 in Equation 5.3.8 gives0 =−(1+∞∑n=012(2n+1)(n+2)2(n+3)C2nz2n+2)· (1+(d−1)F(0,0))−d ·F(0,0).(5.3.12)Solving the above when d = 1 givesG(1,1;z) = 1+∞∑n=012(2n+1)(n+2)2(n+3)C2nz2n+2, (5.3.13)and hence for general d we haveF(0,0) = G(1,d;z) = G(1,1;z)d +(1−d)G(1,1;z) . (5.3.14)In Section 5.4 we will see that the algebraic structure of this solution that givesG(1,d;z) in terms of G(1,1;z) arises naturally from a combinatorial construction.Moreover, this structure extends to the a 6= 1 case.5.3.2 Solution of the Functional Equation When a 6= 1The general a,d case can be solved by the method applied above, however, it issufficient to study the case d = a which can be resolved more cleanly. As men-tioned above the algebraic structure that allows G(1,d;z) to be expressed in terms90of G(1,1;z) extends to give G(a,d;z) in terms of G(a,a;z). We shall see that ex-plicitly in Section 5.4. When d = a the functional Equation 5.2.9 simplifies toK(r,s) ·F(r,s)a2rs = (ar− r− za− zas)as ·F(0,s)− zr2a2 ·F(r,0)+ars.(5.3.15)The symmetries we used above can be reused to remove boundary terms. As abovewe take an alternating sum of transformed equations, but now we must multiplysome of the equations by a non-trivial factor chosen to eliminate boundary terms.The left-hand side becomesLHS = a2rK(r,s)(sF(r,s)− r2sF(r, r2s)+ Lr2s2F(rs, r2s)− Ls2F(rs, 1s)),(5.3.16)whereL = zas−ars+ rs+ zar2zas−ar+ r+ zar2 . (5.3.17)The right-hand side simplifies toRHS = as2(1+ s−a)F(0,s)+a(1+ s−as)F (0,1/s)− (r2 + s)a(s−1)(ar2 +as−2s− s2−1)ar2− r2− s . (5.3.18)Again, we have attempted to massage the functional equation into a form in whichthe coefficients of F(0,s) and F(0,1/s) are independent of r. Unfortunately, wecannot completely clear the denominator of the above functional equation, and wefound it simplest to work with the above expression.Following the method used in the a = 1 case, we can eliminate the left-handside further by choosing a value of r that sets the kernel to 0. We choose the rootwhich gives a positive term power series expansion in z with Laurent polynomial91coefficients in s. Recall that rˆ is given byrˆ(s;z)≡ rˆ = ∑n≥0Cn(1+ s)2n+1z2n+1sn. (5.3.19)Substituting r 7→ rˆ eliminates the left-hand side of the functional equation andwe again consider the coefficient of s1 in the resulting right-hand side. Again,this can be converted into a constant term expression for our generating func-tion. The term as2(1+ s− a)F(0,s) does not contribute to s1. The term a(1+s−as)F (0,1/s) contributes a(1−a)F (0,0). For the remaining term, we considerthe expansion of the expression as a series in a. The coefficient of a1 is[a1]a(rˆ2 + s)(s−1)(arˆ2 +as−2s− s2−1)arˆ2− rˆ2− s = (1− s)(1+ s)2, (5.3.20)and hence[a1s1]a(rˆ2 + s)(s−1)(arˆ2 +as−2s− s2−1)arˆ2− rˆ2− s =−1. (5.3.21)Higher powers of a are (for k ≥ 1)[ak+1]a(rˆ2 + s)(s−1)(arˆ2 +as−2s− s2−1)arˆ2− rˆ2− s=(s−1)(rˆ−1)(rˆ+1)(rˆ− s)(rˆ+ s)rˆ2·(rˆ2s+ rˆ2)k=( (s−1)s2(s+1)2z2 − (s+1)2(s−1))·(rˆ2s+ rˆ2)k=( (s−1)s2(s+1)2z2)·(rˆ2s+ rˆ2)k−((s+1)2(s−1))·(rˆ2s+ rˆ2)k.(5.3.22)To extract the coefficient of s1, we need to consider the expansion of(rˆ2s+rˆ2)kin z.This exponential term simplifies, and we can use Equation 5.3.10d to obtain(rˆ2s+ rˆ2)k= zk(1+ ss)krˆk = ∑p≥0k2p+ k(2p+ kp)(s+1)2p+2ksp+kz2p+2k. (5.3.23)92We will expand the two terms in Equation 5.3.22 individually. For the firstterm, we get[s1] (s−1)s2(s+1)2z2 ·(rˆ2s+ rˆ2)k= ∑p≥0kk+2p(k+2pp)((2k−2+2pk+ p−1)−(2k−2+2pk+ p−2))z2k−2+2p. (5.3.24)We can extract the coefficient of z2n from the above equation by making the sub-stitution n = k−1+ p. We obtain[z2ns1] (s−1)s2(s+1)2z2 ·(rˆ2s+ rˆ2)k= k2n− k+2(2n− k+2n+1)[(2nn)−(2nn−1)].(5.3.25)Therefore[s1] (s−1)s2(s+1)2z2 ·(rˆ2s+ rˆ2)k= ∑n≥k−1k2n− k+2(2n− k+2n+1)[(2nn)−(2nn−1)]z2n.(5.3.26)Following a similar argument for the second term in Equation 5.3.22, we have[s1](s+1)2(s−1) ·(rˆ2s+ rˆ2)k= ∑p≥0kk+2p(k+2pp)((2k+2p+2k+ p)−(2k+2p+2k+ p−1))z2k+2p. (5.3.27)Making the substitution n = k+ p, we get[z2ns1](s+1)2(s−1) ·(rˆ2s+ rˆ2)k= k2n− k(2n− kn)[(2n+2n+1)−(2n+2n)].(5.3.28)93We can then substitute the summation over p with a summation over n.[s1](s+1)2(s−1) ·(rˆ2s+ rˆ2)k= ∑n≥kk2n− k(2n− kn)[(2n+2n+1)−(2n+2n)]z2n.(5.3.29)When n = k−1 in the above equation, the summand reduces to 0 when k > 2. Soit is possible to adjust the range of the summation by adjusting for the k = 1,2cases separately. In those cases, the combined correction terms are a2 and −4a3z2respectively. Thus, we can rewrite[s1](s+1)2(s−1) ·(rˆ2s+ rˆ2)k= ∑n≥k−1k2n− k(2n− kn)[(2n+2n+1)−(2n+2n)]z2n, (5.3.30)with known correction terms for k = 1,2. Combining these summands, we get thatfor n≥ k−1:k2n− k+2(2n− k+2n+1)[(2nn)−(2nn−1)]− k2n− k(2n− kn)[(2n+2n+1)−(2n+2n)]= k(k+1)(2+4n− kn−2k)(k−1−n)(n+1)2(k−2n)(n+2)(2n− kn)(2nn). (5.3.31)Thus taking the coefficient of s1 when r = rˆ in Equation 5.3.18 and accounting forthe correction terms, we get0 = a(a−1)F(0,0)−a+a2−4z2a3−∑k≥1ak+1 ∑n≥k−1k(k+1)(2+4n− kn−2k)(k−1−n)(n+1)2(−2n+ k)(n+2)(2n− kn)(2nn)z2n.(5.3.32)94We exchange the order of summation to give0 = a(a−1)F(0,0)−a+a2−4z2a3−∑n≥0z2nn+1∑k=1k(k+1)(2+4n− kn−2k)(k−1−n)(n+1)2(−2n+ k)(n+2)(2n− kn)(2nn)ak+1. (5.3.33)The extraction of the coefficient [akz2n]F(0,0) requires rearranging the a(a−1)coefficient in front of F(0,0). We express the above equation as:0 = a(a−1)F(0,0)−∑n≥0z2nn+1∑k′=1Qn,k′ak′+1 (5.3.34)for some integers Qn,k′ . This can be rearranged to give:F(0,0) =−(∑n≥0z2nn+1∑k′=1Qn,k′ak′)· 11−a (5.3.35)=−(∑n≥0z2nn+1∑k′=1Qn,k′ak′)·(∑k′′≥0ak′′). (5.3.36)The coefficient of ak from the above is summation of all contributions from k′ andk′′ such that k′+ k′′ = k. Thus:F(0,0) = ∑n≥0z2nn+1∑k=1akk∑k′=1Qn,k′ . (5.3.37)In other words, extracting the coefficient [akz2n]F(0,0) requires a summation of afinite number of the Qn,k′ terms which is obtained from Equation 5.3.33.[akz2n]F(0,0) =k∑k′=0k′(k′+1)(2+4n− k′n−2k′)(k′−1−n)(n+1)2(−2n+ k′)(n+2)(2n− k′n)(2nn)(5.3.38)= k(k+1)(k+2)(2n− k)(n+1)2(n+2)(2n− kn)(2nn). (5.3.39)95We finally obtainF(0,0) = G(a,a) = ∑n≥0z2nn∑k=0akk(k+1)(k+2)(n+1)2(n+2)(2n− k)(2nn)(2n− kn).(5.3.40)This agrees with results due to Brak et al. [9, 54] for a closely related model ob-tained using very different method — we discuss this more fully in the Section 5.4.5.4 Alternate SolutionAn alternate technique for finding the generating function relies on factoring thepairs of walks at each double-visit. First, let us defineG(a,d)≡ G(a,d;z) = F(0,0;a,d;z). (5.4.1)We will frequently hide the z dependence for convenience. Breaking up our con-figurations into pieces between double visits givesG(a,d;z) = 11−dP(a;z) , (5.4.2)where P(a;z) is the generating function of so-called primitive factors. This is quiteanalogous to the classical factorisation of a single Dyck path. These primitive fac-tors are pairs of friendly Dyck-paths which contain no double-visits to the surfaceother than their first and last vertices. Rearranging this expression givesP(a;z) = G(a,d;z)−1dG(a,d;z) =G(a,a;z)−1aG(a,a;z) . (5.4.3)This last expression allows us to calculate P(a;z) from a known expression forG(a,a;z) — such as that given in the previous section. Alternatively, one coulduse results from previous work by Brak et al. [9, 54]. In those works, the authorsconsidered a vesicle model which corresponds exactly to the case d = a — theirvicious walk model can be transformed into the friendly walk model consideredhere, by moving the upper vesicle boundary down by 2 units.Brak et al. use the Lindstro¨m-Gessel-Viennot lemma [26, 38] to express the96partition function of the pair of walks in terms of the partition function of a singlewalk. Namely,[z2n]G(a,a) = S2n(a)S2n+4(a)−S2n+2(a)S2n+2(a)a, (5.4.4)where S2n(a) is the partition function of a single Dyck path of length 2n above awall, and a is conjugate to the number of visits (Equation 3.2.11).S2n(a) = [z2n]2(2−a+a√1−4z2)−1(5.4.5)=n∑k=02k+1n+ k+1(2nn− k)(a−1)k =n∑k=1k2n− k(2n− kn)ak, (5.4.6)where this last formula is taken from [9]. When a = 1 we recover the well-knownCatalan number result, and a well-known central binomial result when a = 2:S2n(1) =Cn =1n+1(2nn), S2n(2) =(2nn). (5.4.7)In light of these simple expressions one can write G(a,a;z) as double sum of prod-ucts of binomials. Using Equation 5.4.3 we write G(a,d;z) in terms of G(a,a;z)G(a,d) = aG(a,a)d +(a−d)G(a,a) , (5.4.8)whereG(a,a) = a−1∞∑n=0[S2n(a)S2n+4(a)−S2n+2(a)S2n+2(a)]z2n, (5.4.9)which simplifies to the expression in Equation 5.3.40 found in the previous section.975.4.1 Solutions at a = 0,1 and 2Since the partition function S2n(a) takes simple values at a = 0,1,2, we haveG(1,1;z) =∞∑n=0[CnCn+2−C2n+1]z2n,=∞∑n=012(2n+1)(n+1)2(n+2)2(n+3)(2nn)2z2n, (5.4.10)G(2,2;z) =∞∑n=0CnCn+1z2n, (5.4.11)andlima→0G(a,a;z)−1a=∞∑n=112(2n−1)n2(n+1)2(n+2)(2n−2n−1)2z2n = z2G(1,1;z). (5.4.12)We can use these together with Equation 5.4.8 to derive expressions for G(1,d)(that agrees with Equation 5.3.13 and Equation 5.3.14) and G(2,d) by simple sub-stitutions. That is,G(1,d) =∑∞n=012(2n+1)(n+1)2(n+2)2(n+3)(2nn)2z2nd +(1−d)∑∞n=0 12(2n+1)(n+1)2(n+2)2(n+3)(2nn)2z2n(5.4.13)andG(2,d) = 2∑∞n=0CnCn+1z2nd +(2−d)∑∞n=0CnCn+1z2n. (5.4.14)A little further work also givesG(0,d) = 11−dz2G(1,1) =11−dz2∑∞n=0 12(2n+1)(n+1)2(n+2)2(n+3)(2nn)2z2n. (5.4.15)This last expression can be derived combinatorially by noting that in the limit a→ 0single visits are forbidden. In this limit, the primitive pieces are in bijection withall walks counted by G(1,1); any primitive piece can be transformed into a pair ofwalks counted by G(1,1) by moving them 1 lattice unit up and gluing edges at the98start and end.5.5 Analysis of Phase Structure and Transitions5.5.1 PhasesWe now turn to the phase diagram of the model which is dictated by the radius ofconvergence of G(a,d;z) as a power series in z. Denote the radius of convergenceby zc(a,d). Equation 5.4.2 shows that the singularities of G(a,d;z) are those ofP(a;z) and the simple pole at 1− dP(a;z) = 0. Denote this latter singularity byzd(a,d). Equation 5.4.3 shows that the singularities of P(a;z) are related to thoseof G(a,a;z) which are known from [9, 54].In particular, the radius of convergence of G(a,a;z) isρ(a) = r.o.c G(a,a;z) =14 = zb a≤ 2,√a−12a = za(a) a > 2.(5.5.1)For a < 2, the thermodynamic phase is related to zb and is the desorbed phase inwhich the walks drift away from the wall and the mean number of visits is O(1).When a > 2, za(a) dominates and the lower walk adsorbs onto the wall and thenumber of visits is O(n). At a = 2, there is a second-order phase transition and ajump discontinuity in the specific heat (the second derivative of the free energy).In both of these phases, the upper walk drifts away from the wall, and the numberof doubly-visited vertices is O(1).In the full a-d model there are 3 phases, two of which are described in theprevious paragraph. In the third phase, associated with the simple pole at zd(a,d),we shall see that the number of doubly-visited vertices is O(n). In what follows, wename these three phases associated with zb,za and zd , desorbed, a-rich and d-richrespectively.995.5.2 Desorbed to a-rich TransitionIn [9], it was shown that the asymptotic behaviour of the singular part of G(a,a;z)near its radius of convergence is given byG(a,a;z)∼A−(1−4z)4 log(1−4z) a < 2,A0(1−4z)2 log(1−4z) a = 2,A+ (1− z/za(a))1/2 a > 2,(5.5.2)where za(a) =√a−12a . It is important to notice that for all a, the singularities areconvergent and therefore G(a,a;z) is convergent on its radius of convergence ρ(a).If we fix d at some small value, and then increase z from 0 towards ρ(a), thenP(a;z) increases from 0 to P(a;ρ(a)). Since d is small, and P(a;ρ(a)) is finite,1−dP(a;ρ(a)> 0 and so the only singularities of G(a,d;z) will be those of P(a;z)and so those of G(a,a;z).Thus for small values of d there is a phase transition on moving a through 2which describes the transition from the desorbed phase to an a-rich phase as occursin [9]. This adsorption transition has been well-studied previously and is unusualin that it has a jump discontinuity in the second derivative of the free-energy ratherthan a divergence.5.5.3 Desorbed to d-rich TransitionLet us restrict our attention to a < 2 and consider the effect of increasing d. Theargument in the previous subsection breaks down as soon as 1−dP(a;ρ(a)) = 0.Call this value dc(a) = P(a;ρ(a))−1.Fix a < 2 and d > dc and consider increasing z from 0 towards ρ(a) = 1/4.The function P(a;z) is an increasing function of z (since it is a positive term powerseries) and so it increases towards P(a;ρ(a)). Since d > dc, P(a;z) will reach thevalue d−1 before it reaches P(a,ρ(a)) and the simple pole will occur when z = zd ,where zd is the solution ofP(a;zd(a,d)) = d−1 (5.5.3)100and zd < zb = 1/4 in this region.Hence, for a < 2, there is a phase transition where zd and zb coincide atdc(a) = P(a;1/4)−1 =aG(a,a;1/4)G(a,a;1/4)−1 . (5.5.4)In order to determine the density of the singly- and doubly-visited vertices in thed > dc phase consider the partial derivatives of zd(a,d) with respect to a and d.Since zd(a,d) is defined by dP(a;zd(a,d)) = 1, the derivatives of zd(a,d) withrespect to a and d are non-zero and so there are positive densities of both singly-and doubly-visited vertices.Now let us turn to the order of this transition; this can be determined by examin-ing the behaviour of P(a;z) close to z = 1/4 which is determined by the behaviourof G(a,a;z) — see Equation 5.4.3. Close to z = 1/4 we can writeG(a,a;z) = Ganalytic(a;z)+Gsingular(a;z), (5.5.5)where the behaviour of Gsingular is given by Equation 5.5.2. Consider an expansionof Ganalytic about z = 1/4Ganalytic(a;z)≈ G(a,a;1/4)+ c1(1−4z)+ · · · (5.5.6)for some non-zero constant c1. The linear correction dominates the dominant sin-gular term in Gsingular. Expanding Equation 5.5.3 about z = 1/4 gives1dc− 1d= P(a;1/4)−P(a;zd),d−dcd2c≈ p1(1−4zd) (5.5.7)for some non-zero constant p1. Hence there is a linear relationship between thelocation of the d-rich singularity, zd , and the distance from the phase boundary.Since the free-energy of the system is − logzd , this also implies the free-energy inthe d-rich phase changes linearly with d. On the other hand in the desorbed phasewhere zb dominates, the free-energy is a constant. From this we see that there is ajump discontinuity in the first derivative of the free-energy and hence this is a first-101order transition. Note that the above argument will also work mutatis mutandis ata = 2.We can observe at finite length a characteristic bimodal probability distributionin the number of doubly-visited vertices — Figure 5.6.0 20 40 60k00.010.020.03Pr(md=k)Figure 5.6: A plot of the probability that a conformation of length 128 has kdoubly-visited sites at a = 1,d = 10.3. This value of d corresponds tothe approximate location of the peak in the specific heat at this length.5.5.4 a-rich to d-rich TransitionThe analysis of the previous section can be adapted to the case a > 2 with someimportant differences. The transition is driven by the singularities za(a) =√a−12aassociated with single-visit adsorption, and the singularity zd(a,d) associated withdouble-visit adsorption. Again, zd(a,d) is the solution of Equation 5.5.3. Thesetwo singularities coincide when d = dc(a) given bydc(a) = P(a;√a−12a)−1=aG(a,a;√a−12a)G(a,a;√a−12a)−1. (5.5.8)102Turning to the order of this transition, we again decompose G(a,a;z) into itsanalytic and singular parts. Observe that close to za(a), Gsingular, given by Equa-tion 5.5.2, dominates the linear part of Ganalytic. Hence we deduce thatdc(a)−d ≈ p2(√a−12a− zd(a,d))1/2, (5.5.9)zd(a,d)≈√a−12a+ p3 (d−dc(a))2 (5.5.10)for some nonzero constants p2, p3. Therefore the free-energy has a jump discon-tinuity in its second derivative on varying d across the transition, and this is asecond-order phase transition. This is very similar to the desorbed to a-rich transi-tion.5.5.5 Phase DiagramWe have established that there are 3 thermodynamic phases; desorbed, a-rich andd-rich. We remind the reader that ma(ϕ) and md(ϕ) denote the number of singleand double visits of ϕ .If we defineA (a,d) = limn→∞〈ma〉nand D(a,d) = limn→∞〈md〉n, (5.5.11)then in the desorbed phase we haveA =D = 0, (5.5.12)while in the a-rich phase we haveA > 0 and D = 0, (5.5.13)and in the d-rich phase has bothA > 0 and D > 0. (5.5.14)In Figure 5.7 and Figure 5.8 we plot 〈ma〉256 and〈md〉256 respectively.103Figure 5.7: A plot of the density of a visits calculated at length n = 256.This highlights the region where it tends to a non-zero constant andcorresponds well to the regions where za and zd dominate. Note thatfor fixed a and increasing, large d we expect that the density of a visitsdecreases though remains positive.Figure 5.8: A plot of the density of d visits calculated at length n = 256. Thishighlights the region where it is tends to a non-zero constant and corre-sponds well to the region where we have shown that zd is the dominantsingularity.104The phase boundary between the desorbed and a-rich phases occurs ata = 2 for d < dc(2). (5.5.15)Note that this phase boundary is, unsurprisingly, independent of d. We can com-pute dc(2) exactly using the results of Section 5.4.1:G(2,2;1/4) = ∑n≥0CnCn+116−n = 8− 643pi ; (5.5.16)dc(2) =2G(2,2)G(2,2)−1 =16(8−3pi)64−21pi ≈ 11.55159579. (5.5.17)In a similar way we can compute dc(0) and dc(1):dc(0) =30pi165pi−512 ≈ 14.81234030; (5.5.18)dc(1) =8(512−165pi)4096−1305pi ≈ 13.47187382. (5.5.19)The transitions to the d-rich phase from the desorbed and a-rich phases aregiven bydc(a) = P(a;ρ(a))−1 =aG(a,a;ρ(a))G(a,a;ρ(a))−1 , (5.5.20)where ρ(a) is given by Equation 5.5.1. We plot P(a;ρ(a)) and G(a,a;ρ(a)) inFigure 5.9.In the limit as a→ ∞, dc(a)→ 2a+ o(a). As a→ ∞, the generating functionG(a,a) is dominated by those configurations which have a maximal number ofvisits. In this case, the lower walk simply zig-zags along the wall and the upperwalk is effectively unconstrained by the lower. Hencelima→∞G(a,a;z) = ∑n≥0Cnanz2n = 1−√1−4az22az2. (5.5.21)Substituting z = ρ(a) then gives G(a,a;ρ(a)) = 2+ o(1) from which the asymp-105totics of dc(a) follows:dc(a)∼ 2a as a→ ∞. (5.5.22)In the next section we compute the above asymptotic form in more detail.0 10 20a012G(a,a;ρ(a))0 10 20a00.050.1P(a;ρ(a))Figure 5.9: A plot of both G(a,a;ρ(a)) and P(a;ρ(a)). The dotted line in-dicates a = 2. In the plot of G(a,a;ρ(a)) we have also marked theasymptotic form computed below.Combining all of this information gives the phase diagram for our model,which we present in Figure 5.10. It is interesting to note that the three transi-tion lines meet with the two critical lines forming an angle. Classically this wouldindicate mean-field like behaviour of a bicritical point. If true, this mean-fieldbehaviour would be interesting to understand.106a-richd-richdesorbed0 5 10a01020dFigure 5.10: The phase diagram of our model. The three phases are as indi-cated and the first-order transition is marked with a dashed line, whilethe two second-order transitions are marked with solid lines. The threeboundaries meet at the point (a,d) = (2,11.55 . . .).5.5.6 Asymptotics of the d-rich- a-rich Phase BoundaryWe now consider how the d-rich a-rich phase boundary, dc(a), behaves for large ain more detail. From Equation 5.3.40, we write [z2nak]G(a,a;z) in closed form[z2nak]G(a,a;z) = k(k+1)(k+2)(n+1)2(n+2)(2n− k)(2nn)(2n− kn). (5.5.23)We seek the asymptotic form of G(a,a;ρ(a)) as a→∞, and so we need to evaluatethe asymptotics ofG(a,a;ρ(a)) = ∑n≥0(a−1)n(n+1)2(n+2)4na2n(2nn) n∑k=0k(k+1)(k+2)(2n− k)(2n− kn)ak.(5.5.24)107Expanding this slightly further givesG(a,a;ρ(a)) = ∑n≥01(n+1)2(n+2)4na2n(2nn)n∑j,k=0(−1) jk(k+1)(k+2)(2n− k)(nj)(2n− kn)a j+k. (5.5.25)Substitute j = `− k to getG(a,a;ρ(a)) = ∑n≥01(n+1)2(n+2)4n(2nn) 2n∑`=0a`−2nn∑k=0(−1)`−kk(k+1)(k+2)(2n− k)(n`− k)(2n− kn). (5.5.26)So now the coefficient of a0 is[a0] =∞∑n=01n+1(2nn)4−n = 2, (5.5.27)while[a−1] =−∞∑n=13n(n+1)(n+2)(2nn)4−n =−2, (5.5.28)[a−2] = 0 (5.5.29)[a−3] =∞∑n=22(n−1)(n+1)(n+2)(2nn)4−n = 1, (5.5.30)and[a−4] =∞∑n=33(n−2)(n+1)(n+2)(2nn)4−n = 54. (5.5.31)These simple forms continue as far as we have observed. This then givesG(a,a;ρ(a))∼ 2− 2a+ 0a2+ 1a3+ 54a4+ 1516a5+ 732a6+O(a−7). (5.5.32)108We can then plot this against our numerical estimates of G(a,a;ρ(a)). One shouldnote that the series G(a,a,ρ(a)) converges very slowly for a > 2. Since we knowthe summands decay as n−3/2, we can assume that the partial sums, sn, grow asA+Bn−1/2. We can then accelerate the convergence of the series by estimating Awith the sequenceAn = sn(n+√n√n−1)− sn−1(n−1+√n√n−1). (5.5.33)This combination was chosen by solving the pair of simultaneous equations sn =A+Bn−1/2,sn−1 = A+B(n− 1)−1/2. We found that sequence An converged farfaster than the partial sums sn.5.6 Discussion5.6.1 Nature of SolutionIn Section 5.5, we demonstrated that when a = 1, the model undergoes a phasetransition at d = dc(1) = 8(512−165pi)4096−1305pi . Since this is not an algebraic number, it fol-lows that the generating function of the model does not satisfy a linear differentialequation in z with integer polynomial coefficients in a,d and z. That is, it cannotbe D-finite.Consider, to the contrary, that the generating functionG(1,d;z) = [r0s0] f (r,s;1,d;z) (5.6.1)is a D-finite power series in z with integer polynomial coefficients in d. By defini-tion, it satisfies a non-trivial linear differential equation of the formpk(d;z)∂ kG∂ zk + . . . p1(d;z)∂G∂ z + p0G(1,d;z) = 0, (5.6.2)where the p j(d;z) are integer polynomials in d and z. By standard results in thetheory of linear differential equations, the singularities of G(1,d;z) are zeros of theleading polynomial pk(d;z).For small d we know that the dominant singularity of G(1,d) is zb = 1/4. At109the critical value of d, there is a change in dominant singularity from zb to zd .Exactly at the critical value, zb = zd = 1/4. Thus the discriminant of pk(d;z) withrespect to d must be zero at this point. Since the discriminant is a polynomial inz,d with integer coefficients and z = 1/4, it follows that this critical value of dmust be an algebraic number. Above we showed that dc(1) = 8(512−165pi)4096−1305pi which isnot algebraic and thus G(1,d;z) is not D-finite. A standard result [39] on D-finiteseries states that specialisation of D-finite series are themselves D-finite and thusG(a,d;z) cannot be D-finite and nor is f (r,s;a,d;z).The Lindstro¨m-Gessel-Viennot lemma combines the partition functions of single-walk models — equivalent to sums and Hadamard products of the underlyingsingle-walk generating function which is algebraic (this is true quite generally —see [6]). Any finite combination of Hadamard products and sums of algebraic orD-finite generating functions remains D-finite [39] and thus the Lindstro¨m-Gessel-Viennot lemma (alone) cannot be applied to decompose the model considered hereinto single-walk problems.That being said, the Lindstro¨m-Gessel-Viennot lemma can be combined with afactorisation argument to yield a solution as we demonstrated in Section 5.4.5.6.2 Fixed Energy Ratio Models: r-modelsFinally, let us now consider the family of physical models parameterised by −∞<r < ∞ whereεd = rεa and so d = ar (5.6.3)that allows us to summarise our results. Let us call these r-models.For any r-model the high temperature phase is the desorbed state. The modeleffectively already analysed by Brak et al. [9] and Owczarek et al. [54] has d = aand so r = 1. In this case there is a single low temperature phase being the a-richphase. Given there are no additional phase boundaries for a < d one can deducethat for all r ≤ 1 the model goes from the desorbed state at high temperaturesthrough a single second-order phase transition to the a-rich phase at low tempera-tures.The special point in our phase diagram where (a,d) = (2,11.55 . . .) where the110three phases meet occurs in the r-model withr = rt ≡log(11.55 . . .)log2= 3.53 . . . . (5.6.4)For all r ≥ rt there is a single low temperature phase which is the d-rich phase: thetransition on lowering the temperature is now first-order.Since we have shown above that dc(a)∼ 2a as a→ ∞ one can now argue thatfor all 1< r < rt the r-model has two phase transitions on lowering the temperature.At very low temperatures the model is in a d-rich phase while at high temperaturesthe model is in the desorbed state. At intermediate temperatures the system isin an a-rich phase. Both transitions, from desorbed to a-rich, and a-rich to d-rich, are second-order transitions with jump discontinuities in the specific heat.In Figure 5.11 we plot the fluctuations in a-visits as a function of temperature atlength 128 for the r = 2 r-model: two peaks occur in these fluctuations.0 5 10a02040〈m2 a〉−〈ma〉2Figure 5.11: A plot of the fluctuations in the number of a-visits, ma, forlength n = 128 as function of a clearly showing two peaks.If one argues that a physically realisable model is one where both walks pick111up the same energy when they touch the surface together then the model is theone with r = 2, that is d = a2. It is interesting to see that this model containstwo phase transitions: one at a = 2 and the other at a ≈ 3.301 found by solvingEquation 5.5.20 for dc = a2c . In any case, we have a family of adsorption modelsthat have one or two low temperature states and which the order of the transitionchanges as the parameter is varied. We have analysed this model using an exactsolution and fully delineated its behaviour. It will be of interest to analyse thebehaviour of this model in a slit.112Chapter 6Two Directed Walks in the SlitWhen we confine a ring polymer between two sticky walls, the situation becomesmore complex. This situation has been studied by various directed and non-directedlattice walk models [1, 10, 11, 27, 46, 51, 70]. Here the phase diagram of the modelcan depend on the mesoscopic size of the polymer relative to the width of the slab/s-lit and the strengths of the interactions on both walls. A motivation for studyingthis type of system is related to modelling the stabilisation of colloidal dispersionsby adsorbed polymers (steric stabilisation) and the destabilisation when the poly-mer can adsorb on surfaces of different colloidal particles (sensitised flocculation).A polymer confined between two parallel plates exerts a repulsive force on theconfining plates because of the loss of configurational entropy unless the polymeris attracted to both walls when it can exert an effective attractive force at largedistances.A directed walk model of a polymer confined between two sticky walls wasstudied by Brak et al. [10] and rederived in Chapter 4. Let us now briefly reviewthe findings of that work so as to motivate the model we study in this paper. In theirmodel the polymers are represented by Dyck paths, which are directed paths in theplane, taking north-east and south-east steps starting on, ending on and stayingabove the horizontal axis. These are classical objects in combinatorics [23]. Theheight of these paths is then restricted; this is interpreted as a model of a polymerconfined between two walls that are w lattice units apart, as in Figure 6.1. It will becrucial to understand the results to note that the polymer is attached to the bottom113wall at its end. Finally, different Boltzmann weights a and b were added for eachvisit to the bottom and top walls respectively. The partition function for these pathsFigure 6.1: A Dyck path confined between two walls spaced w lattice unitsapart. Each visit to the bottom wall contributes a Boltzmann weight aand each visit to the top wall contributes a Boltzmann weight b. Forcombinatorial reasons we do not weight the first vertex.is defined asZsinglen (a,b;w) = ∑ϕ∈S nwama(ϕ)bmb(ϕ) , (6.0.1)where S nw is the set of Dyck paths of length n of restricted height with maximumw, ma(ϕ) the number of vertices on the bottom wall and mb(ϕ) the number ofvertices on the top wall (excluding the leftmost vertex).A phase transition can only occur when both the thermodynamic limit andthe limit of infinite width (to give a two-dimensional thermodynamic system) aretaken. However, it was explained by Brak et al. [10] that taking the thermodynamiclimit n→∞ before or after taking the width of the slit to infinity is crucially impor-tant. If the width, w, of the system is taken to infinity first then the walk does notsee the top wall and a half plane system is retrieved, since the polymer is tetheredto the bottom wall. There is a simple adsorption transition as a is varied: a secondorder phase transition occurs when a = 2. On the other hand, if the thermodynamiclimit is taken before the width is taken to infinity then a different phase diagramensues dependent on both a and b.We define the reduced free energy κsingle(a,b;w) for single Dyck paths at fixed114finite w asκsingle(a,b;w) = limn→∞1nlogZsinglen (a,b;w) (6.0.2)and taking the limit w→ ∞ gives the so-called infinite slit limit:κsinglein f−slit(a,b)≡ limw→∞κsingle(a,b;w) = limw→∞limn→∞1nlogZsinglen (a,b;w) . (6.0.3)This limit is different from the half-plane limitκsinglehal f−plane(a) = limn→∞ limw→∞1nlogZsinglen (a,b;w) =log(2) if a≤ 2log(a√a−1)if a > 2,(6.0.4)which is independent of b.It was shown in [10] (and in Chapter 4) thatκsinglein f−slit(a,b) =log(2) if a,b≤ 2log(a√a−1)if a > 2 and a > blog(b√b−1)otherwise.(6.0.5)For small a and b the walk is desorbed from both walls, while the large a and bphases are characterised by the order parameter of the thermodynamic density ofvisits to the bottom and top walls respectively. Correspondingly, there are 3 phasetransition lines. The first two are given by b = 2 for 0 ≤ a ≤ 2 and a = 2 for0≤ b≤ 2. These lines separate the desorbed phase from the two adsorbed phasesand are lines of second order transitions of the same nature as the one found inthe half-plane model. There is also a first order transition for a = b > 2 wherethe density of visits to each of the walls jumps discontinuously on crossing theboundary non-tangentially (see Figure 6.2 (left) and Figure 4.4).For finite widths the effective force between the walls, induced by the polymer,115was defined [10] asF (a,b;w) = κ(a,b;w)−κ(a,b;w−1) . (6.0.6)For large w it was found that the sign and length scale of the force depended onthe values of a and b and that it was more refined that simply following the phasediagram (see Figure 6.2 (right) and Figure 4.5).Figure 6.2: (left) Phase diagram of the infinite strip for a single walk. Thereare three phases: desorbed, adsorbed onto the bottom wall (ads bottom)and adsorbed onto the top (ads top). (right) A diagram of the regions ofdifferent types of effective force between the walls of a slit for a singleDyck path. Short range behaviour refers to exponential decay of theforce with slit width while long range refers to a power law decay. Thezero force curve is given by ab = a+ b. On the dashed line there is asingular change of behaviour of the force.The regions of the plane which gave different asymptotic expressions for κand hence different phases for the infinite slit clearly also give different force be-haviours. For the square 0 ≤ a,b ≤ 2 the force is repulsive and long-range as itdecays as a power law while outside this square the force decays exponentially andso is short-ranged. This change coincides with the phase boundary of the infiniteslit phase diagram. However, the special curve ab = a+ b is a line of zero forceacross which the force, while short-ranged on either side (except at (a,b) = (2,2)),changes sign. Hence this curve separates regions where the force is attractive (tothe right of the curve) and repulsive to the left of the curve. The line a = b for a > 2is also special and, while the force is always short-ranged and attractive, the range116of the force on the line is discontinuous and twice the size on this line than closeby. All these features leads to a force diagram that encapsulates these features (seeFigure 6.2 (right)). It should be recalled here that the behaviour of the directedsystem described above has been shown to be a faithful representation of the moregeneral undirected self-avoiding walk model [46, 70].It is not unreasonable to speculate that the inequality of the infinite-slit and halfplane limits, and more generally the resultant force diagram may be dependenton the particular single walk model chosen where the polymer was tethered tothe bottom wall. There is no natural single walk model with fixed ends that cancircumvent this restriction sensibly. One is therefore led to consider models ofmultiple walks in a slit where walks can be tethered to both walls. In fact a relatedgeneralisation has already been considered by Alvarez et al. [1] where they studieda model of self-avoiding polygons confined to a slit. The resulting force diagramis quite different from the single-walk diagram shown in Figure 6.2 (left).In this paper we consider a directed walk model of two polymers confinedbetween two walls with which the polymers interact, as in the single polymer modeldescribed above. In particular we fully analyse the infinite slit phase diagram andthe large width force behaviour as a function of the interaction parameters. Weshow there are distinct differences from the single walk problem.6.1 ModelWe consider pairs of directed paths of equal length in a width w strip of the squarelattice — namely Z×{0,1, . . . ,w}, taking steps (1,±1). These paths may touch(ie share edges and vertices) but not cross — such walks are often referred toas friendly. There is a body of literature on systems of vicious walks (the nameoriginates in [22]) that may neither cross nor share edges. We consider those pairsof friendly paths whose initial vertices lie at at (0,0) and (0,w). We note here thatour system of directed friendly walks can be mapped to a similar system of viciouswalks by translating the top walk up by 2 lattice units.Let ϕ be such a pair of paths and define |ϕ| to be the length of the paths.If the width of the strip, w, is odd then the paths never share vertices and thecombinatorics that follows is more complicated. Because of this we only consider117Figure 6.3: Two walks confined between two walls spaced w lattice unitsapart. Each visit of the bottom walk to the bottom wall contributes aBoltzmann weight a and each visit of the top walk to the top wall con-tributes a Boltzmann weight b. For combinatorial reasons we do notweight the leftmost vertex of either walk.even widths. Note that this implies that the distance between the endpoints of thepaths is always even.To complete our model let we add the energies −εa and −εb for each visit ofthe walks to the bottom and top walls respectively (aside from the leftmost vertexof each walk). The number of visits of the bottom walk to the bottom walk willbe denoted ma(ϕ) while the number of visits of the top walk to the top wall willbe denoted mb(ϕ) — again excluding the leftmost vertex of each walk. The mainmodel we discuss in the paper is based on pairs of walks, ϕ , that finish withendpoints together at the same height. Define the corresponding partition functionto beZn(a,b;w) =∑ϕe(εama(ϕ)+εbmb(ϕ))/kBT =∑ϕama(ϕ)bmb(ϕ) , (6.1.1)where T is the temperature, kB the Boltzmann constant and a = eεa/kBT and b =eεb/kBT are the Boltzmann weights associated with visits. The thermodynamic re-duced free energy at finite width is given in the usual fashion asκ(a,b;w) = limn→∞1nlog(Zn(w)) . (6.1.2)Because the model at finite w is essentially one-dimensional, the free energy118is an analytic function of a and b and no thermodynamic phase transitions occur[36]. As noted above, the infinite slit limit for the single walk model does displaysingular behaviour and so we consider the same limit for this model. The infiniteslit free energy for the two walk model is found analogously byκin f−slit(a,b) = limw→∞κ(a,b;w) = limw→∞limn→∞1nlogZn(a,b;w). (6.1.3)Motivated by the single walk problem, we see that the above quantity could bedifferent when the order of limits is swapped. Since we have defined the modelso that walks start on opposite walls, when the width is taken to infinity before thelength, the system separates into two half-planes. Consequently we refer to thislimit as the double half-plane limit and so defineκdouble−hal f−plane(a,b) = limn→∞limw→∞1nlogZn(a,b;w). (6.1.4)Since the system separates into two half-planes we haveκdouble−hal f−plane(a,b) = κsinglehal f−plane(a)+κsinglehal f−plane(b). (6.1.5)Motivated by the single walk model, we consider the effective force applied tothe walls by the polymersFn =1n[log(Zn(w))− log(Zn(w−2))] , (6.1.6)with a thermodynamic limit ofF (a,b;w) = κ(a,b;w)−κ(a,b;w−2). (6.1.7)Note that we will consider only systems of even width and hence we had to modifythe single walk definition.Given that the double half-plane limit is known from the discussion above, weshall concentrate on the infinite slit limit. In this limit, the free energy does notdepend on where the walks end. It turns out that the combinatorics of the modelin which the walks end together are easier. Accordingly we study the generating119functionG(a,b;z) =∞∑n=0Zn(w)zn. (6.1.8)where the partition function now counts only those walks which end together. Theradius of convergence of the generating function zc(a,b;w) is directly related to thefree energy viaκ(a,b;w) =− log(zc(a,b;w)) . (6.1.9)Below we tackle the enumeration of our friendly walkers problem through theformalism of generating functions and the kernel method. As noted above, we cantransform our system into a system of vicious walkers by translating the top walk2 lattice units up. One could then study the partition function of the sytem of 2walks in terms of a determinant of single-walk systems via the Lindstro¨m-Gessel-Viennot lemma [26, 34, 38]. Since we chose to focus on asymptotics, which aremore easily obtained from generating functions, we did not take that approach.6.2 Functional EquationsFigure 6.4: We form the generating function of all pairs of paths that start inboth surfaces and end anywhere according to their length, and distancesof the endpoints from the surfaces. The path depicted contributes z9r1s1to the generating function.Though we are primarily interested in the behaviour of pairs of paths that sharetheir final vertices, we will need to define the generating function of more generalpairs of paths with no restrictions on their endpoints. Define du(ϕ) to be the dis-120tance from the endpoint of the upper path to the top of the strip. Similarly defined`(ϕ) to be the distance of the endpoint of the lower path to the bottom of the strip.6.2.1 Without InteractionsLet us first consider the case when a = b = 1. We construct the generating functionF(r,s;z)≡ F(r,s) = ∑ϕ∈pathsz|ϕ|rd`(ϕ)sdu(ϕ), (6.2.1)where r,s are conjugate to the distances of the endpoints to either boundary and zis conjugate to length. See Figure 6.4. In order to construct a functional equationsatisfied by this generating function we also need to define the generating functionof those paths whose final vertices touch.rwFd(s/r;z)≡ rwFd(s/r) =w∑h=0shrw−h ·[shrw−h]{F(s,r)} (6.2.2)where we have used[sirk]{F(s,r)} to denote the coefficient of sirk in the generat-ing function F(s,r). The generating function G(1,1;z) = Fd(1;z). Also note thatsince the problem is vertically symmetric, we have F(r,s)≡F(s,r) and rwFd(s/r)=swFd(r/s). Further note that[sirk]{F(s,r)} is zero whenever i− k is not even.One can construct all pairs of paths using a column-by-column constructionwhose details we give below. Translating the construction into its action on thegenerating functions gives the following functional equationF(r,s) = 1+ z(s+ 1s)(r+ 1r)·F(r,s)− zr(s+ 1s)·F(0,s)− zs(r+ 1r)·F(r,0)+ zsr·F(0,0)− zsr · swFd(r/s). (6.2.3)We now explain each of the terms in this equation. The trivial pair of paths consistsof two isolated vertices at (0,0) and (0,w). This gives the initial 1 in the right-handside of the above functional equation. Note that to enumerate directed polygons inthe strip we may replace the above with all pairs of vertices lying at the same121vertical ordinate; this would replace 1 with ∑wk=0 rksw−k = (sw+1− rw+1)/(s− r).Figure 6.5: Every pair of paths can be continued by appending directed stepsto their endpoints as shown. While there are at most 4 possible combi-nations, depending on the distance from boundaries, some combinationswill be forbidden.See Figure 6.5. When the endpoints are away from the boundaries, every pair ofpaths may be continued by appending directed steps in four different ways. Sinceeach of these steps either increases or decreases the distance of the endpoint fromthe boundary the result isz(s+ 1s)(r+ 1r)·F(r,s). (6.2.4)Figure 6.6: When the endpoints of the walks are close to the boundaries onemust take care to subtract off the contributions of the configurations thatstep outside the strip as depicted here.See Figure 6.6. When the endpoints are close to the boundaries or each other,then appending steps as described above may result in paths that either step outsidethe strip or cross each other. If the endpoint of the upper path lies on the boundary122then one cannot append a (1,1) step to that path. Such configurations are countedbyzs(r+ 1r)·[s0]F(r,s)≡ zs(r+ 1r)F(r,0). (6.2.5)Similarly if the endpoint of the lower path lies on the boundary then one cannotappend a (1,−1) step to that path:zr(s+ 1s)·[r0]F(r,s)≡ zr(s+ 1s)F(0,s). (6.2.6)Figure 6.7: (left) When removing the contributions of paths that step out-side the strip, we over-correct by twice removing those configurationsin which both paths step outside the strip simultaneously. (right) Whenthe endpoints of the paths are close together we must remove the con-tribution of paths that cross each other.We correct the enumeration by subtracting both of these contributions. In sodoing we over-correct by subtracting twice the contribution of paths whose end-points lie on opposite boundaries (see Figure 6.7(left)). Thus we add back inzsr·[s0r0]F(r,s)≡ zsrF(0,0) (6.2.7)Finally, we must also remove the contribution of those paths whose endpointscross. This happens when we take a path whose endpoints lie together and attemptto append an upward step to the lower path and a downward step to the upper path123(see Figure 6.7(right)). So we must subtractzsr · rwFd(s/r)≡ zsr · swFd(r/s), (6.2.8)where this equivalence comes from the vertical symmetry of the model withoutinteractions.6.2.2 Interacting ModelWe now add boundary interactions to this model. We weight each pair of pathsaccording to the number of contacts the upper (lower) path has with the upper(lower) boundary excluding their leftmost vertices. Recall that a is conjugate tothe number of contacts between the lower path and the boundary and similarly bis conjugate to the number of contacts between the upper path and the boundary.Thus our generating functions F and Fd become functions of a,b in addition tor,s,z; as above we will typically write these asF(r,s;a,b;z)≡ F(r,s) and Fd(s/r;a,b;z)≡ Fd(s/r). (6.2.9)Figure 6.8: Interactions with the boundary are produced when one or bothpaths steps from distance one onto the boundary.We now modify the above construction by noting that a contact between theupper path and its boundary is created when an upward step is appended to a path124lying 1 step from the boundary (see Figure 6.8). Thus we addzb(r+ 1r)[s1]{F(r,s)} . (6.2.10)However these configurations have already been enumerated with incorrect weight,so we must also subtractz(r+ 1r)[s1]{F(r,s)} . (6.2.11)Thus we arrive atz(b−1)(r+ 1r)[s1]{F(r,s)} . (6.2.12)And similarly, by considering contacts between the lower path and the lower bound-ary we obtainz(a−1)(s+ 1s)[r1]{F(r,s)} . (6.2.13)Again we find that these terms over-correct and we must consider those configu-rations in which contacts with the upper and lower boundaries are created at thesame time.z(a−1)(b−1)[s1r1]{F(r,s)} . (6.2.14)So finally we have the following functional equation for F(r,s;a,b;z)≡ F(r,s).F(r,s) =1+ z(s+ 1s)(r+ 1r)·F(r,s)− zr(s+ 1s)·F(0,s)− zs(r+ 1r)·F(r,0)+ zsr·F(0,0)− zsr · swFd(r/s)+ z(b−1)(r+ 1r)[s1]{F(r,s)}+ z(a−1)(s+ 1s)[r1]{F(r,s)}+ z(a−1)(b−1)[s1r1]{F(r,s)} . (6.2.15)We can now further simplify this equation by rewriting[s1]{F(r,s)} ,[r1]{F(r,s)}125and[s1r1]{F(r,s)} in terms of F(r,0),F(0,s) and F(0,0).Extracting the coefficient of s0r0 in the above equation givesF(0,0) =1+ z(1+(b−1)+(a−1)+(a−1)(b−1))[s1r1]{F(r,s)}=1+ zab[s1r1]{F(s,r)} . (6.2.16)This has a simple combinatorial interpretation; any path with endpoints ending ineach surface must either be trivial or can be constructed from a shorter path whoseendpoints end a single unit from each boundary.Similarly, extracting the coefficient of s0 in the above givesF(r,0) = 1+ z(r+ 1r)[s1]{F(r,s)}− zr[s1r0]{F(s,r)}+ z(b−1)(r+ 1r)[s1]{F(r,s)}+ z(a−1)[r1s1]{F(r,s)}+ z(a−1)(b−1)[s1r1]{F(r,s)}= 1+ zb(r+ 1r)[s1]{F(r,s)}+ zb(a−1)[s1r1]{F(r,s)} (6.2.17)and similarlyF(0,s) = 1+ za(r+ 1r)[r1]{F(r,s)}+ za(b−1)[s1r1]{F(r,s)} . (6.2.18)This gives three linear equations and we may solve them to obtainz[s1r1]{F(r,s)}= F(0,0)−1ab(6.2.19)z(s+ 1s)[r1]{F(r,s)}=−1+F(r,0)+(b−1)F(0,0)ab(6.2.20)z(r+ 1r)[s1]{F(r,s)}=−1−F(0,s)+(a−1)F(0,0)ab. (6.2.21)126Substituting these into the original interactions equation gives usF(r,s) = 1ab+ z(s+ 1s)(r+ 1r)·F(r,s)− zsr · swFd(r/s)+A(r,s)F(0,s)+B(r,s)F(r,0)+C(r,s)F(0,0), (6.2.22)whereA(r,s) = 1− 1b− z(r+1/r)s,B(r,s) = 1− 1a− z(s+1/s)r,C(r,s) = zsr−(1− 1a)(1− 1b). (6.2.23)From this equation we can recover G(a,b;z) = Fd(1;a,b;z). In the following sec-tion we do not solve explicitly for Fd , however we are able to determine its singu-larities and so its asymptotic behaviour.6.3 Solution of Functional EquationsAt this point, we define v = w/2 as is the more natural parameter in what follows.Rather than solving the full model directly, we first examine the special cases ofa = b = 1 and a = b.6.3.1 Without InteractionsWe start by collecting the F(r,s) terms in Equation 6.2.3 to get(1− z(s+ 1s)(r+ 1r))F(r,s) = 1− zr(s+ 1s)·F(0,s)− zs(r+ 1r)·F(r,0)+ zsr·F(0,0)− zsr · s2vFd(r/s). (6.3.1)The coefficient of F(r,s) is called the kernel K(r,s;z)≡ K(r,s) and its symme-127tries play a key role in the solutionK(r,s) = 1− z(s+ 1s)(r+ 1r). (6.3.2)We use the kernel method which exploits the symmetries of the kernel to re-move boundary terms in the functional equation (see [8] for a thorough descriptionof the kernel method). The kernel is symmetric under the following operations(r,s) 7→(1r,s)(r,s) 7→(r, 1s)(r,s) 7→ (s,r) . (6.3.3)To more be precise, we use the above symmetries to construct the following equa-tionsK(r,s) ·F(r,s) = 1− zs(r+ 1r)F(r,0)− zr(s+ 1s)F(0,s)+ zsrF(0,0)− zrs2v+1Fd( rs)(6.3.4a)K(1r,s)·F(1r,s)= 1− zs(r+ 1r)F(1r,0)− zr(s+ 1s)F(0,s)+ zrsF(0,0)− zs2v+1rFd(1rs)(6.3.4b)K(r, 1s)·F(r, 1s)= 1− zs(r+ 1r)F (r,0)− zr(s+ 1s)F(0, 1s)+ zsrF(0,0)− zrs2v+1Fd (rs)(6.3.4c)K(1r, 1s)·F(1r, 1s)= 1− zs(r+ 1r)F(1r,0)− zr(s+ 1s)F(0, 1s)+ zrsF(0,0)− zrs2v+1Fd( sr).(6.3.4d)We can eliminate the boundary terms by taking the appropriate alternating sumof the above equations:rs · (Equation 6.3.4a)− sr· (Equation 6.3.4b)− rs· (Equation 6.3.4c)+ 1rs· (Equation 6.3.4d) . (6.3.5)This is similar to the “orbit-sum” discussed in [7, 8] as well as Section 5.3.128Since the kernel is the same in all of the above equations we obtainK(r,s) · (Sum of F) = (s−1)(s+1)(r−1)(r+1)rs+ zs2v+2r2Fd(1rs)+ zr2s2v+2Fd (rs)− zs2v+2r2Fd(rs)− zr2s2v+2Fd( sr). (6.3.6)The symmetry of Fd described by Equation 6.2.8 comes from the vertical sym-metry of the model; it can be extended to givezr2v+1sFd( sr)≡ zrs2v+1Fd(rs)zr2v+1s2v+1Fd(1rs)≡ Fd(rs). (6.3.7)These relations will then simplify the functional equation further:K(r,s) · (Sum of F) = (s−1)(s+1)(r−1)(r+1)rs+ z(r2v+4 + s2v+4)s2v+2r2v+2Fd (rs)−z(r2v+4s2v+4 +1)r2v+2s2Fd(rs). (6.3.8)We can now remove the left hand side of the equation by choosing values ofr and s that set the kernel to zero. That is, K(rˆ, sˆ) = 0; this also gives z−1 =(sˆ+ 1sˆ)(rˆ+ 1rˆ). Making this substitution gives0 = (sˆ−1)(sˆ+1)(rˆ−1)(rˆ+1)rˆsˆ+(rˆ2v+4 + sˆ2v+4)sˆ2v+1rˆ2v+1 (rˆ2 +1)(sˆ2 +1)Fd (rˆsˆ)−(rˆ2v+4sˆ2v+4 +1)rˆ2v+1sˆ(rˆ2 +1)(sˆ2 +1)Fd(rˆsˆ). (6.3.9)By eliminating denominators, we obtain0 = (sˆ−1)(sˆ+1)(rˆ−1)(rˆ+1)(rˆ2 +1)(sˆ2 +1)rˆ2vsˆ2v+(rˆ2v+4 + sˆ2v+4)Fd (rˆsˆ)− sˆ2v(rˆ2v+4sˆ2v+4 +1)Fd(rˆsˆ). (6.3.10)We now apply a similar argument used by Bousquet-Me´lou [7] to determine129the singularities of Fd . Set rˆ = qsˆ for a root of unity q 6=−1 such that q2v+4 =−1.More precisely, we choose sˆ as a solution to K(qs,s) = 0. The above equation thenreduces to0 = (sˆ4q4−1)(sˆ4−1)(sˆq)2vsˆ2v+ sˆ2v+4(q2v+4 +1)Fd(qsˆ2)− sˆ2v(q2v+4sˆ4v+8 +1)Fd (q) . (6.3.11)Since q2v+4 =−1, the second term drops out and we can find an explicit equationfor Fd(x) at the roots of unity q.Fd(q) =(sˆ4q4−1)(sˆ4−1)(sˆq)2v1− sˆ4v+8 . (6.3.12)We can see that Fd must be a rational function of z since it can be translatedinto a problem of counting paths via a finite transfer matrix (see, for example,Chapter V of [23] and Section 2.2).The construction of Fd(x) ensures that it is a polynomial in x of degree 2v.Thus, we can obtain the full Fd(x) by using Lagrange polynomial interpolation andthe known points of Fd(q) (we follow the method in [7]). By taking a set of {qk}such that q2v+4k =−1 with qi 6=−1 for any i and making the substitutions, we getFd(x) =2v∑j=0Fd(q j) ∏0≤m≤2vm6= jx−qmq j−qm. (6.3.13)Note that no term in the product contributes any singularities in z. Thus Fd(x)being singular implies at least one Fd(qk) is also singular. By Equation 6.3.12,we can see that Fd(q) will be singular when sˆ (and hence rˆ) is a (4v+ 8)-th rootof unity. Combining with the kernel, a superset of singularities can obtained byvarious choices of k, j:z j,k =1(rˆ+ 1rˆ)(sˆ+ 1sˆ) = 14cos(pi j2v+4)cos( pik2v+4) . (6.3.14)Note that since rˆ = qsˆ with q2v+4 =−1, we do not have j = k in the above and so130the dominant singularity is obtained when j = 1,k = 2 (or vice-versa).6.3.2 With Equal Interactions a = bFor this section, we follow the same argument however the details become morecomplicated due to the boundary terms. We start by arranging Equation 6.2.22 tocollect all F(r,s) terms to obtain the equationK(r,s)F(r,s) = 1ab− zsr · s2vFd(r/s)+A(r,s)F(0,s)+B(r,s)F(r,0)+C(r,s)F(0,0), (6.3.15)whereA(r,s) = 1− 1a− z(r+1/r)s,B(r,s) = 1− 1a− z(s+1/s)r,C(r,s) = zsr−(1− 1a)2(6.3.16)with the kernelK(r,s) = 1− z(r+ 1r)(s+ 1s). (6.3.17)Since the kernel is the same as that of the non-interacting case we can use thesame symmetries and combine the four equations to eliminate the boundary termsF(r,0),F(1r ,0),F(0,s) and F(0, 1s). This results in the following functional equa-131tionK(r,s) · (linear combination of F) =rs2v+1(s2−1)(r2−1)(a−1)2(r2s2z+ r2z− sr+ s2z+ z)z ·F(0,0)+(sza+ r2sza+ r− ra)(rsa− rs− s2za− za)zs4v+3 ·Fd(1rs)− (rsa− rs− s2za− za)(za+ r2za− rsa+ rs)z ·Fd( sr)+(sza+ r2sza+ r− ra)(rs2za+ rza+ s− sa)zr3s4v+3 ·Fd(rs)− (za+ r2za− rsa+ rs)(rs2za+ rza+ s− sa)zr3 ·Fd (rs)− rs2v+1z2(s4−1)(r4−1). (6.3.18)Since the wall interaction is symmetric, we can again make use of the verticalsymmetry to eliminate Fd(1rs)and Fd(sr)and giveK(r,s) · (linear combination of F)= L(r,s;a) ·F(0,0)+M(r,s;a) ·Fd(rs)+N(r,s;a) ·Fd (rs)− rs2v+1z2(s4−1)(r4−1), (6.3.19)where L(r,s;a),M(r,s;a) and N(r,s;a) are easily computed though complicatedfunctions. As before, we pick values of r and s that set the kernel to 0. And sinceK(rˆ, sˆ) = 0 we can write z = (sˆ+ 1/sˆ)−1(rˆ + 1/rˆ)−1 and so eliminate it from thecoefficients of the above equations. After clearing the denominators, we obtaina functional equation with coefficients α,β and δ (again being easily computed,though complicated, functions)0 = α(r,s;a) ·Fd(rˆsˆ)+β (r,s;a) ·Fd (rˆs)+δ (r,s;a). (6.3.20)The coefficient δ is important in what follows, and so we state it explicitlyδ (r,s;a) = r2vs2v(1− r4)(1− s4). (6.3.21)132Note that if r or s are fourth roots of unity then δ = 0.Unlike the a = b = 1 case, there is no simple relation between rˆ and sˆ that willgive us an explicit form for Fd(x). However, we can still extract the location of thesingularities by solving when the coefficients α and β are simultaneously 0 withδ 6= 0. Solving α = β = 0, we getrˆ2v = rˆ2(a−1)−1rˆ2(a−1− rˆ2) sˆ2v =− sˆ2(a−1)−1sˆ2(a−1− sˆ2) (6.3.22)orrˆ2v =− rˆ2(a−1)−1rˆ2(a−1− rˆ2) sˆ2v = sˆ2(a−1)−1sˆ2(a−1− sˆ2) . (6.3.23)Since the form of rˆ and sˆ is similar, we will concentrate on finding the solutions ofrˆ2v = rˆ2(a−1)−1rˆ2(a−1− rˆ2) (6.3.24)and from there, deducing solutions for sˆ.By rearranging the equation, we geta−1 =rˆv+2− 1rˆv+2rˆv− 1rˆv. (6.3.25)Since a is a positive real parameter, the right hand side must also be real. Thefollowing theorem tells us that all solutions to this equation must lie either on theunit circle or the real line.Theorem 6.1. The expressionrˆv+2− 1rˆv+2rˆv− 1rˆv(6.3.26)is real if and only if rˆ ∈ R or if |rˆ|= 1. The equivalent statement holds for sˆ.The proof of this statement is given in the Appendix A and is relatively straight-forward though cumbersome.We can further refine Theorem 6.1 when a≤ 2 and in that case all the solutionslie on the unit circle. To do this, we use of Theorem 2.14.133By rearranging Equation 6.3.22, we get0 = rˆ2v+2(a−1− rˆ2)− (rˆ2(a−1)−1) (6.3.27)0 = sˆ2v+2(a−1− sˆ2)+(sˆ2(a−1)−1) (6.3.28)which is in the form given in the theorem with n = 2, h(z) = (a− 1− z2) andλ =±1. The zeros of h(z) are given byz =±√a−1. (6.3.29)Hence, the zeros of h(z) will be inside the closed disc exactly when a ≤ 2 and sowe can apply the theorem.We note that when rˆ and sˆ lie on the unit circle, the singularities of the gener-ating function are of a similar form to that given in Equation 6.3.14. However, theangles are not simple functions of w. In Section 6.4.4 we give asymptotic expres-sions for the singularities.6.3.3 With Interactions, a, b FreeWe proceed via the same argument as per the previous sections. We start by ar-ranging Equation 6.2.22 to collect all F(r,s) terms to obtain the equationK(r,s)F(r,s) = 1ab− zsr · s2vFd(r/s)+A(r,s)F(0,s)+B(r,s)F(r,0)+C(r,s)F(0,0), (6.3.30)whereA(r,s) = 1− 1b− z(r+1/r)s,B(r,s) = 1− 1a− z(s+1/s)r,C(r,s) = zsr−(1− 1a)(1− 1b)(6.3.31)with the same kernel as before.134Again, use the symmetries of the kernel to construct 4 linear equations and thentake linear combinations to eliminate the boundary terms F(r,0),F(1r ,0), F(0,s)and F(0, 1s). This results in the following functional equationK(r,s) · (linear combination of F)= rs2v+1(s2−1)(r2−1)(a−1)(b−1)(r2s2z+ r2z− sr+ s2z+ z)z ·F(0,0)+(szb+ r2szb+ r− rb)(rsa− rs− s2za− za)zs4v+3 ·Fd(1rs)− (rsa− rs− s2za− za)(zb+ r2zb− rsb+ rs)z ·Fd( sr)+(szb+ r2szb+ r− rb)(rs2za+ rza+ s− sa)zr3s4v+3 ·Fd(rs)− (zb+ r2zb− rsb+ rs)(rs2za+ rza+ s− sa)zr3 ·Fd (rs)− rs2v+1z2(s4−1)(r4−1). (6.3.32)Unlike the previous case, the wall interactions are no longer symmetric andhence we cannot apply the vertical symmetry. However, we can pick values rˆ andsˆ that sets the kernel K(rˆ, sˆ) = 0 and eliminate z from the equation. Making thissubstitution, we get0 = sˆ4v+2(b−1− sˆ2)(rˆ2(a−1)−1) ·Fd(1rˆsˆ)− (1− sˆ2(b−1))(rˆ2(a−1)−1) ·Fd(sˆrˆ)− sˆ4v+2rˆ2(b−1− sˆ2)(a−1− rˆ2) ·Fd(rˆsˆ)+ rˆ2(sˆ2(b−1)−1)(a−1− rˆ2) ·Fd(rˆsˆ)− sˆ2v(sˆ4−1)(rˆ4−1). (6.3.33)Up to this point, we have omitted the dependence of the parameters a and b inFd(x) for convenience. In full detail, Fd(x) ≡ Fd(x;a,b). This will be importantin the next step when we look at the result of mapping a↔ b. For this, we defineGd(x) = Fd(x;b,a).135With a little work we haveGd(x) = Fd(x;b,a) (6.3.34)= x2vFd(1x;a,b). (6.3.35)Swapping a↔ b in Equation 6.3.33, we get0 = sˆ4v+2(a−1− sˆ2)(rˆ2(b−1)−1) ·Gd(1rˆsˆ)− (1− sˆ2(a−1))(rˆ2(b−1)−1) ·Gd(sˆrˆ)− sˆ4v+2rˆ2(a−1− sˆ2)(b−1− rˆ2) ·Gd(rˆsˆ)+ rˆ2(sˆ2(a−1)−1)(b−1− rˆ2) ·Gd(rˆsˆ)− sˆ2v(sˆ4−1)(rˆ4−1). (6.3.36)Now convert Gd back to Fd using the relation Gd(x) = x2vFd(1x)and clear denom-inators to find0 = rˆ4v+2(sˆ2(a−1)−1)(b−1− rˆ2) ·Fd(1rˆsˆ)− rˆ4v+2s2(a−1− sˆ2)(b−1− rˆ2) ·Fd(sˆrˆ)− (sˆ2(a−1)−1)(rˆ2(b−1)−1) ·Fd(rˆsˆ)+ sˆ2(rˆ2(b−1)−1)(a−1− sˆ2) ·Fd(rˆsˆ)− rˆ2v(sˆ4−1)(rˆ4−1). (6.3.37)Combining Equation 6.3.33 and Equation 6.3.37, we can eliminate one moreboundary term (e.g. Fd(1rˆsˆ)) resulting in0 = α(rˆ, sˆ) ·Fd(rˆsˆ)+β (rˆ, sˆ) ·Fd(sˆrˆ)+ γ(rˆ, sˆ) ·Fd (rˆsˆ)+δ (rˆ, sˆ). (6.3.38)We do not state all of the coefficients α,β ,γ (they are easily computed but compli-136cated), however the coefficient δ will be important in what followsδ = r2vs2v(1− r4)(1− s4)[(1−b+ r2)(1+ s2−as2)r2v+2− (1−b+ s2)(1+ r2−ar2)s2v+2]. (6.3.39)Note that for a,b in this general case, δ = 0 when r,s are fourth roots of unity orr = s.We follow the same logic as for the previous section. The locations of thesingularities are when the functions α = β = γ = 0 and δ 6= 0. Thus, solving forwhen α = β = γ = 0 simultaneously givesr4v+4 = (r2(b−1)−1)(r2(a−1)−1)(b−1− r2)(a−1− r2) and s4v+4 = (s2(b−1)−1)(s2(a−1)−1)(b−1− s2)(a−1− s2) .(6.3.40)By rearranging Equation 6.3.40, we get0 = rˆ4v+4((b−1− rˆ2)(a−1− rˆ2))−((rˆ2(b−1)−1)(rˆ2(a−1)−1))(6.3.41)0 = sˆ4v+4((b−1− sˆ2)(a−1− sˆ2))−((sˆ2(b−1)−1)(sˆ2(a−1)−1))(6.3.42)which is in the form given in the Theorem 2.14 with n = 4, h(z) = (b−1− z2)(a−1− z2) and λ = 1. The zeros of h(z) are given byz =±√a−1,±√b−1. (6.3.43)Hence, the zeros of h(z) will be inside the closed disc exactly when a,b≤ 2. Con-sequently when a,b ≤ 2 we know that rˆ, sˆ lie on the unit circle. When a or b > 2we observe that all the solutions lie either on the unit circle or the real line.6.4 Exact and Asymptotic ResultsIn this section, we will describe the asymptotic and exact results we obtained foreach of the cases. In the case where both a,b ∈ {1,2} or ab = a+b, we are able toobtain an exact solution for the dominant singularity. However, more generally weare only able to obtain asymptotic results. Note that by a↔ b symmetry, we need137only consider cases where a ≥ b. This gives 13 different cases (see Figure 6.9)which we summarise in Section 6.4.14.Figure 6.9: The a−b parameter space contains 13 representative points, de-pending on whether a,b = 1, 1 < a,b < 2, a,b = 2, a,b > 2, or if a = bor if a,b lie on along a special curve ab = a+ b. The numbers in thisdiagram correspond to the cases described in the text.In what follows we proceed by solving Equation 6.3.40 for possible values ofrˆ, sˆ; we are able to do this exactly for a small number of cases, but in the majoritywe must do so asymptotically. Each pair of rˆ, sˆ may lead to a singularity of thegenerating function however only when the auxiliary function δ is non-zero.6.4.1 Case (I) : a = b = 1.This case is the non-interacting case. We can obtain the asymptotic expansion bylooking at Equation 6.3.14 with j = 1,k = 2.zc =14+ 532pi2v2− 58pi2v3+O(v−4). (6.4.1)1386.4.2 Case (II): a = b = 2.Simplifying the solutions for rˆ and sˆ in Equation 6.3.22, we get thatrˆ4v = 1rˆ4sˆ4v = 1sˆ4. (6.4.2)This suggests that the solutions for rˆ and sˆ are simple roots of unity. Hence the setof solutions given byrˆ ∈{exp[pii j2v+2]}0≤ j≤4v+4sˆ ∈{exp[piik2v+2]}0≤k≤4v+4(6.4.3)is a superset of singularities for rˆ and sˆ.If we attempt to set both rˆ, sˆ = 1, we do not obtain a valid singularity sinceδ = 0. To obtain the dominant singularity, we instead takerˆ = exp[pii2v+2]sˆ = 1, (6.4.4)and with this choice δ 6= 0. Note that by symmetry we could also swap the choicesof rˆ↔ sˆ. We then havezc =1(rˆ+ 1rˆ)(sˆ+ 1sˆ) = 14cos( pi2v+2)= 14+ 132pi2v2− 116pi2v3+O(v−4). (6.4.5)6.4.3 Case (III): a = 2; b = 1.As per the previous two cases, we find that the particular choice of a and b leads tosolutions that are roots of unity. Equation 6.3.40 reduces torˆ4v =− 1rˆ6sˆ4v =− 1sˆ6, (6.4.6)139and so the solutions are given byrˆ ={exp[pii j4v+6]}0≤ j≤4v+4j oddsˆ ={exp[piik4v+6]}0≤k≤4v+4k odd. (6.4.7)To obtain the dominant singularity we take j,k = 1,3 respectively:rˆ = exp[pii4v+6]sˆ = exp[3pii4v+6], (6.4.8)and this gives a non-zero δδ = −6pi3v3+ 27pi3v4+ pi3(47pi2−1296)16v5+O(v−6). (6.4.9)The dominant singularity iszc =14cos( pi4v+6)cos(3pi4v+6) , (6.4.10)and its asymptotic expansion iszc =14+ 564pi2v2− 1564pi2v3+O(v−4). (6.4.11)Note that if we tried choosing j,k = 1,1 then rˆ = sˆ and δ = 0.6.4.4 Case (IV): a = b; a < 2.In Case (I) and Case (II), the solutions of rˆ and sˆ are simply roots of unity. Hencewe guess that for this generalised case 1< a= b< 2, the solutions of rˆ and sˆ will beperturbations of the roots of unity found in the a = b = 1 case (a similar approachwas used in [10]). More precisely, we look for a solution of the formrˆ = exp[ipiv+2(c0 +c1v+ c2v2+ · · ·)], (6.4.12)140and similarly for sˆ. We substitute this into Equation 6.3.22 and solve for the un-known constants. This process yieldedrˆ = exp[ipiv− 2a−2(1− 4a(a−1)pi23(v(a−2)−1)3 +O(1(v(a−2)−2)5))](6.4.13)which, when substituted into Equation 6.3.22 givesrˆ2v− rˆ2(a−1)−1rˆ2(a−1− rˆ2) =8ia(a−1)(a2 +8a−8)pi515(a−2)4v5 +O(v−6). (6.4.14)Repeating this for sˆ leads tosˆ = exp[ipi2(v− 2a−2)(1− a(a−1)pi23(−2+ v(a−2))3 +O(1(−2+ v(a−2))5))](6.4.15)which, when substituted into Equation 6.3.22 givessˆ2v + sˆ2(a−1)−1sˆ2(a−1− sˆ2) =ia(a−1)(a2 +8a−8)pi560(a−2)4v5 +O(v−6). (6.4.16)Note that Equation 6.3.22 is not symmetric under rˆ↔ sˆ.This choice of rˆ and sˆ gives a δ value ofδ = rˆ2vsˆ2v(rˆ4−1)(sˆ4−1) = 8pi2v2+ 8ipi2(3api−4i)(a−2)v3 +O(v−4)(6.4.17)which is non-zero. Hence, using solving the kernel equation K(rˆ, sˆ) = 0 for z, weget thatzc =14+ 532pi2v2+ 58pi2v3(a−2) +O(v−4). (6.4.18)We see that as a→ 1 this agrees with Case (I).6.4.5 Case (V): a = b; a > 2.In the case a > 2, Theorem 2.14 does not hold and we expect Equation 6.3.22 tocontain extra solutions along the real axis. By rearranging Equation 6.3.22, we get141that(a−1− rˆ2)rˆ2v+2 = rˆ2(a−1)−1, (6.4.19)(a−1− sˆ2)sˆ2v+2 =−sˆ2(a−1)−1. (6.4.20)We observe that rˆ =√a−1 will set the left hand side to zero and leave a smallremainder on the right. Hence, we looked at solutions that perturb this square root(again a similar approach was used in [10]). We proceed as per the previous caseand arrive at a solution of the formrˆ =√a−1[1− a(a−2)2(a−1)2(a−1)v +O(v(a−1)−2v)](6.4.21)sˆ = 1√a−1[1+ a(a−2)2(a−1)2(a−1)v +O(v(a−1)−2v)]. (6.4.22)This choice of rˆ and sˆ will give a non-zero δ which to leading order isδ = (a−1)2va2(a−2)2 +O(v) . (6.4.23)Putting this together with the kernel equation K(rˆ, sˆ) = 0 we getzc =a−1a2+ (a−2)2a2(a−1)(a−1)v +O(v(a−1)−2v). (6.4.24)6.4.6 Case (VI): a < 2; b < 1.In Case (I), Case (II) and Case (IV), the solutions of rˆ and sˆ are simple perturbationsof roots of unity. Hence we guess that for the case 1 < a,b < 2, the solutions of rˆand sˆ will be of a similar nature. Hence we apply a similar method to that used in142Case (IV) but now applied to Equation 6.3.40. This leads us torˆ = exppii(v− a+b−4(a−2)(b−2))1− 2(ab−a−b)(a2b+ab2−10ab+8a+8b−8)pi23(a−2)3(b−2)3(v− a+b−4(a−2)(b−2))3+O((v− a+b−4(a−2)(b−2))−5))]; (6.4.25)which, when substituted into Equation 6.3.40 givesrˆ4v+4− (rˆ2(b−1)−1)(rˆ2(a−1)−1)(b−1− rˆ2)(a−1− rˆ2) = O(1(a−2)5(b−2)5v5). (6.4.26)We remind the reader that in this case if rˆ = sˆ then δ = 0 and so we need the valueof sˆ to be different. Following the same trend as for the previous case, we get thatsˆ= exppii2(v− a+b−4(a−2)(b−2))1− (ab−a−b)(a2b+ab2−10ab+8a+8b−8)pi26(a−2)3(b−2)3(v− a+b−4(a−2)(b−2))3+O((v− a+b−4(a−2)(b−2))−5))]; (6.4.27)which, when substituted into Equation 6.3.40 givessˆ4v+4− (sˆ2(b−1)−1)(sˆ2(a−1)−1)(b−1− sˆ2)(a−1− sˆ2) = O(1(a−2)5(b−2)5v5). (6.4.28)This choice of rˆ and sˆ will give a δ value ofδ =−16pi2(a−2)(b−2)v2− 16ipi2(6abpi−6bpi−2bi−2ai−9api+6pi+8i)v3+O(v−4)(6.4.29)which is non-zero. Hence, solving the kernel equation K(rˆ, sˆ) = 0 for z, we get thatzc =14+ 532pi2v2+ 516pi2(a+b−4)v3(a−2)(b−2) +O(v−4). (6.4.30)143Note that Equation 6.4.30 reduces to Equation 6.4.18 when b = a, and reducesto Equation 6.4.1 when a,b→ 1.6.4.7 Case (VII): a > 2; b > 2.In the case where a or b is greater than 2, we argue as for Case (V) in that we expectsolutions along the real axis as well. Since rˆ and sˆ satisfy the same equation and theequation is invariant under switching a and b, we can (without loss of generality)look at the expansion of rˆ in terms of√a−1. We getrˆ =√a−1[1+ a(ab−a−b)(a−2)2(a−1)3(a−b)(a−1)2v +O((a−1)−4v)]. (6.4.31)Using the same process, we get thatsˆ =√b−1[1+ b(ab−a−b)(b−2)2(b−1)3(b−a)(b−1)2v +O((b−1)−4v)]. (6.4.32)We then check that this gives a non-zero value of δ . For simplicity of notation, welet A = a−1 and B = b−1 and through abuse of notation, we obtainδ = A2vBv[A(AB−1)(A−B)(A2−1)(B2−1)+O(A−2v)+O(B−v)]. (6.4.33)By making the substitution into K(rˆ, sˆ) = 0, we get that to leading orderzc =√a−1√b−1ab+ (a−2)2(ab−a−b)√b−12ab(b−a)√a−1(a−1)2v+2+ (b−2)2(ab−a−b)√a−12ab(a−b)√b−1(b−1)2v+2+O(a−4v)+O(b−4v). (6.4.34)Note that the above expression implies that zc is a decreasing function of v. Tosee this, consider a > b > 2. The first correction term is now negative (since(b−a)< 0) while the second correction term is positive. The factor of (a−1)2v+2in the denominator of the first correction term is larger than the corresponding fac-tor of (b−1)2v+2 in the denominator of the second term. Hence for large v the firstcorrection term is smaller and negative than the larger and positive second correc-tion term. Finally as v→ ∞ the sum of two corrections is positive and shrinking to1440.6.4.8 Case (VIII): a > 2; b < 1.The next region we consider is when one parameter is small (< 2) and the other islarge (> 2). Without loss of generality, we can assume that a > 2 and b < 2. Wemake use of the solutions obtained in Case (VI) and Case (VII) to obtainrˆ =√a−1[1+ a(ab−a−b)(a−2)2(a−1)3(a−b)(a−1)2v +O((a−1)−4v)](6.4.35)andsˆ= exppii2(v− a+b−4(a−2)(b−2))1− (ab−a−b)(ab2 +a2b−10ab+8a+8b−8)pi26(a−2)3(b−2)3(v− a+b−4(a−2)(b−2))3+O((v− a+b−4(a−2)(b−2))−5))]. (6.4.36)Substituting these choices into δ give the following non-zero formδ = 2piA2v(AB−1)(A2−1)[− i(A−1)v+O(v−2)]. (6.4.37)We can then extract the growth rate aszc =√a−12a[1+ pi28v2+ pi2(a+b−4)4(a−2)(b−2)v3 +O(v−4)]. (6.4.38)Note that as b→ 1 the above expression becomes Equation 6.4.48 in Case (X)below.We now complete the analysis by looking at the remaining boundary cases.1456.4.9 Case (IX): a < 2; b = 1.In this case, Equation 6.3.40 reduces down torˆ4v+6 = rˆ2(a−1)−1a−1− rˆ2 sˆ4v+6 = sˆ2(a−1)−1a−1− sˆ2 . (6.4.39)Following similar techniques used Case (VI), we can obtain the two primitiveroots of rˆ and sˆ to getrˆ = exp[pii2(v+ a−3a−2)(1− a(a−1)pi26(a−3+ v(a−2))3+O(1(a−3+ v(a−2))5))](6.4.40)andsˆ = exp[pii(v+ a−3a−2)(1− 2a(a−1)pi23(a−3+ v(a−2))3+O(1(a−3+ v(a−2))5))].(6.4.41)Using these values of rˆ and sˆ, we obtain a non-zero δ value ofδ =−16pi2(a−2)v2− 16pi2i(2ia+3pia−6i)v3+O(v−4). (6.4.42)This will yield a dominant singularity ofzc =14+ 532pi2v2− 516pi2(a−3)(a−2)v3 +O(v−4). (6.4.43)As a→ 1 this reduces to Equation 6.4.1.6.4.10 Case (X): a > 2; b = 1.In this case, we have the same equations for rˆ and sˆ as the previous caserˆ4v+6 = rˆ2(a−1)−1a−1− rˆ2 sˆ4v+6 = sˆ2(a−1)−1a−1− sˆ2 . (6.4.44)146Following methods used in Case (V) and Case (VII), we can obtain the singularityof rˆ on the real line:rˆ =√a−1[1− a(a−2)2(a−1)4(a−1)v +O(v(a−1)−2v)](6.4.45)whilesˆ = exp[pii2(v+ a−3a−2)(1− a(a−1)pi26(a−3+ v(a−2))3+O(1(a−3+ v(a−2))5))].(6.4.46)Using these values of rˆ and sˆ, we obtain a non-zero δδ = 2iapi(a−1)2v+2(a−2)2v+O((a−1)2vv2). (6.4.47)This yields a dominant singularity aszc =√a−12a[1+ pi28v2− (a−3)pi24(a−2)v3 +O(v−4)]. (6.4.48)6.4.11 Case (XI): a = 2; b < 2.This case is very similar to that of Case (IX). Equation 6.3.40 reduces down torˆ4v+4 =− rˆ2(b−1)−1b−1− rˆ2 sˆ4v+4 =− sˆ2(b−1)−1b−1− sˆ2 . (6.4.49)Again we follow the method used in Case (VI), and we findrˆ = exp[pii4(v+ b−4b−2)(1− b(b−1)pi23(b−4+2v(b−2))3+O(1(b−4+2v(b−2))5))](6.4.50)sˆ = exp[3pii4(v+ b−4b−2)(1− 3b(b−1)pi2(b−4+2v(b−2))3+O(1(b−4+2v(b−2))5))].(6.4.51)147These give a non-zero δ :δ = 6pi3(b−2)v3− 3ipi3(3ib+6pib−12i−4pi)v4+O(v−5). (6.4.52)And so we find the dominant singularity:zc =14+ 564pi2v2− 564pi2(b−4)(b−2)v3 +O(v−4). (6.4.53)Note that as b→ 1 we recover Equation 6.4.11.6.4.12 Case (XII): a > 2; b = 2.As per Case (XI), we assume that b = 2. This reduces Equation 6.3.40sˆ4v+4 =− sˆ2(a−1)−1a−1− sˆ2 . (6.4.54)Looking at the expansion of sˆ, we getsˆ = exp[pii2(2v− a−4a−2)(1− pi2(a−1)a3((2a−4)v+a−4)3+O(1((2a−4)v+a−4)5))](6.4.55)Similarly, the solution for rˆ is given by a simplified version of Equation 6.4.31.rˆ =√a−1[1+ a(a−2)2(a−1)3(a−1)2v +O((a−1)−4v)]. (6.4.56)Together these giveδ = (a−1)2v[pia(a−1)(a−2)3v+O(v−2)](6.4.57)with the dominant singularity beingzc =√a−12a[1+ pi232v2− (a−4)pi232(a−2)v3 +O(v−4)]. (6.4.58)1486.4.13 Case (XIII): ab−a−b = 0.Looking at Case (VI), Case (VII) and Case (VIII), the factor ab−a−b appears inthe asymptotic expansions, leading us to believe that there may be something ofinterest along this line. We note that this polynomial plays an important role in thesingle-walk version of this model [10] — along the curve ab = a+b the dominantsingularity is independent of the width of the system. While this is not the case forthe two-walk model we consider in this paper, we are able to compute the dominantsingularity exactly along the curve.Equation 6.3.40 reduces down to(rˆ2(a−1)−1)(a−1− rˆ2)(rˆ2v+2−1)(rˆ2v+2 +1) = 0, (6.4.59)(sˆ2(a−1)−1)(a−1− sˆ2)(sˆ2v+2−1)(sˆ2v+2 +1) = 0. (6.4.60)This suggests that the solutions of rˆ or sˆ come in two forms. One is a simpleroot of unity and the other is a square root type singularity. Again, the conditionδ 6= 0 requires rˆ 6= sˆ and we obtain the following exact expressionsrˆ =√a−1 (6.4.61)sˆ = exp(pii2v+2). (6.4.62)We could equally well have chosen the above with rˆ and sˆ swapped. Using theabove values of rˆ and sˆ, we obtainδ = (a−1)2v[−2ia2(a−2)3piv+ 2(−2i+ ia−pi+pia)pi(a−2)2a2v2+O(v−3)].(6.4.63)This will yield a dominant singularity ofzc =√a−12acos( pi2v+2) (6.4.64)or asymptotically,zc =√a−12a[1+ pi28v2− pi24v3+O(v−4)]. (6.4.65)149Note that as a→ 2 this reduces to Equation 6.4.5.6.4.14 SummaryHere we simply summarise the results of this section and divided them into threetables. In Table 6.1 we give the cases in which we are able to find the dominantsingularity exactly. For the remainder of the parameter space we have been unableto find exact expressions and we present only asymptotic results. These are dividedinto Table 6.2 and Table 6.3 according to whether or not at least one a,b exceeds2. For comparison we include the asymptotics of the single-walk model with b = 1in Table 6.4.Case: a b Dominant Singularity (zc)(I) = 1 = 1 = 14cos( pi2v+4)cos( 2pi2v+4)= 14 + 532 pi2v2 −58pi2v3 +O(v−4)(II) = 2 = 2 = 14cos( pi2v+2)= 14 + 132 pi2v2 −116pi2v3 +O(v−4)(III) = 2 = 1 = 14cos( pi4v+6)cos( 3pi4v+6)= 14 + 564 pi2v2 −1564pi2v3 +O(v−4)(XIII) ab = a+b =√a−12acos( pi2v+2)=√a−12a(1+ pi28v2 −pi24v3 +O(v−4))Table 6.1: The exact value and asymptotic behaviour of the dominant singu-larity when a,b ∈ 1,2 and ab = a+b. Note that in each case zc decreaseswith increasing v.6.5 Overview and Discussion6.5.1 Infinite Slit Phase DiagramRecall that in the single walk case, discussed in the introduction, the order of thelimits polymer length n and slit width w going to infinity matters; it was shown in150Case: a b Dominant Singularity (zc)(IV) a = b < 2 = 14 + 532 pi2v2 +58pi2v3(a−2) +O(v−4)(VI) < 2 < 2 = 14 + 532 pi2v2 +516pi2(a+b−4)v3(a−2)(b−2) +O(v−4)(IX) < 2 = 1 = 14 + 532 pi2v2 −516pi2(a−3)v3(a−2) +O(v−4)(XI) = 2 < 2 = 14 + 564 pi2v2 −564pi2(b−4)v3(b−2) +O(v−4)Table 6.2: The asymptotic behaviour of the dominant singularity when a,b≤2. Again note that in each case, zc is a decreasing function of v and thatzc→ 14 as v→ ∞.Case: a b Dominant Singularity (zc)(V) a = b > 2 = a−1a2 +(a−2)2a2(a−1)(a−1)v +O(v(a−1)2v)(VII) > 2 > 2 =√a−1√b−1ab+ (a−2)2(ab−a−b)√b−12ab(b−a)√a−1(a−1)2v+2 +(b−2)2(ab−a−b)√a−12ab(a−b)√b−1(b−1)2v+2+O(a−4v)+O(b−4v)(VIII) > 2 < 2 =√a−12a[1+ 18 pi2v2 +14pi2(a+b−4)(a−2)(b−2)v3 +O(v−4)](X) > 2 = 1 =√a−12a[1+ 18 pi2v2 −14pi2(a−3)(a−2)v3 +O(v−4)](XII) > 2 = 2 =√a−12a[1+ 132 pi2v2 −132pi2(a−4)(a−2)v3 +O(v−4)]Table 6.3: The asymptotic behaviour of the dominant singularity when atleast one of a,b > 2. Note that zc decreases with increasing v in all cases.[10] thatκsinglehal f−plane(a) 6= κsinglein f−slit(a,b). (6.5.1)In fact the phase diagram for the single walk in the infinite slit, given in Fig-ure 6.2(left), depends on both a and b whereas the half plane limit depends onlyon a. This can be understood by observing that a finite Dyck path must visit thebottom wall as it is fixed at both ends there so once the width is sent to infinity anyfinite Dyck path only feels the bottom wall, while if the length of the Dyck path isfirst sent to infinity the walk will “see” both walls for any finite width.151a zc Asymptotic expansion1 12cos( pi2v+2) ∼12 + pi216v2 −pi28v3 +O(v−4)(1,2) ◦ ∼ 12 + pi216v2 −pi28(2−a)v3 +O(v−4)2 12cos( pi4v+2) ∼12 + pi264v2 −pi264v3 +O(v−4)(2,∞) ◦ ∼√a−1a(1+ (a−2)22(a−1)2v+2)+O((a−1)−4v)Table 6.4: The dominant singularity when b = 1 for the single-walk model.From the calculations in the previous section we see that for the two walkmodel the infinite slit free energy isκin f−slit(a,b) =log(4) if a,b≤ 2log(2a√a−1)if a > 2 and b < 2log(2b√b−1)if a < 2 and b > 2log(ab√a−1√b−1)if a≥ 2 and b≥ 2.(6.5.2)Hence the phase diagram can be illustrated as in Figure 6.10.We observe thatκin f−slit(a,b) = κsinglehal f−plane(a)+κsinglehal f−plane(b) (6.5.3)and recalling Equation 6.1.5 we see thatκin f−slit(a,b) = κdouble−hal f−plane(a,b). (6.5.4)So the free energy for this two walk model does not depend on the order of thelimits.This conclusion depends on the particular model we have chosen where bothwalks start on different walls. Had we considered a model where both walks startedon the bottom wall this observation would be different; by taking the width toinfinity first, neither walk would interact with the top wall and the free energy of152Figure 6.10: Phase diagram of the infinite strip for the two walk model anal-ysed in this paper. There are four phases: a desorbed phase, a phasewhere the bottom walk is adsorbed onto the bottom wall, a phase wherethe top walk is adsorbed onto the top wall, and a phase where bothwalks are adsorbed onto their respective walls.this system would be that of two walks in a single half-plane. On the other hand, theinfinite slit free energy does not depend on the end points of the polymer becausethe length is taken to infinity first.6.5.2 Force Between the WallsUsing the asymptotic expressions for κ found above we obtain the asymptotics forthe force. We have• For a,b < 2F ∼ 5pi2w3; (6.5.5)• For a < 2,b = 2F ∼ 5pi22w3; (6.5.6)• For a < 2,b > 2F ∼ pi2w3; (6.5.7)153• For a > 2,b < 2F ∼ pi2w3; (6.5.8)• For a = 2,b < 2F ∼ 5pi22w3; (6.5.9)• For a = 2,b = 2F ∼ pi2w3; (6.5.10)• For a > 2,b = 2F ∼ pi24w3; (6.5.11)• For a = 2,b > 2F ∼ pi24w3; (6.5.12)• For a,b > 2 with a > bF ∼ (b−2)2(ab−a−b) log(b−1)2(a−b)(b−1)3(1b−1)w; (6.5.13)• For a,b > 2 with a < bF ∼ (a−2)2(ab−a−b) log(a−1)2(b−a)(a−1)3(1a−1)w; (6.5.14)• For b = a > 2F ∼ (a−2)2 log(a−1)2(a−1)2(1a−1)w/2. (6.5.15)For any a,b the force is positive and so is repulsive. This is in contrast to thesingle walk case where there is a region of attractive forces. The regions of theplane which gave different asymptotic expressions for κ and hence different phasesfor the infinite slit clearly also give different force behaviours. There is also aspecial subtle change of the magnitude of the force when a = b for a,b > 2. On the154other hand the special super-integrable curve a+ b = ab does not display specialbehaviour, which relates to which walk is less bound to its respective surface andso drives the value of the force, except when a = b = 2.The difference between the single and two walk models can be understoodas follows. When there are two walks they effective shield each other from theinteractions of the other wall and it is when a single walk is sufficiently attractedto the two sides of the slit simultaneously that an attractive force eventuates. Thereare however changes in the magnitude and the range of the repulsive force arisingfrom whether the walks are adsorbed or desorbed. When either or both walks aredesorbed there is a long range force arising from the entropy of the walk(s) whileif both are adsorbed the force is short-ranged as the excursions of either walk fromthe walls are relatively short-ranged. The force diagram is given in Figure 6.11.Figure 6.11: A diagram of the regions of different types of effective forcebetween the walls of a slit. Short range behaviour refers to exponentialdecay of the force with slit width while long range refers to a powerlaw decay. On full lines there is a change from long to short range forcedecay. On the dashed lines there is a singular change of behaviour ofthe magnitude of the force.1556.5.3 ConclusionA model of two polymers confined to be in a long macroscopic sized slit withsticky walls has been modelled by a directed walk system. Our results show dis-tinct differences from the earlier single polymer results. In particular, we see dif-ferences from the single polymer system in both the phase diagram, and the signand strength of the entropic force exerted by the polymers on the walls of the slit.The phase diagram contains four phases, whereas that the single walk modelhas only three. Moreover, this phase diagram is independent on the order oneconsiders the limits of large width and length to be taken. This is also in contrastwith the single walk system.The force induced by the polymers remains repulsive in all parts of the phasediagram even though the range of the force does depend on whether the walks areadsorbed onto the walls. This again is in contrast with the single polymer systemwhere an attractive regime is observed. In our two polymer system each polymeris effectively shielded from the opposite wall by the other polymer. This givesrise to the difference between the results seen here and those of the single polymersystem.While we have a model that goes beyond the single polymer results, to obtaina situation which might replicate the non-directed self-avoiding polygon results ofAlvarez et al. [1] one will need to allow both walks to interact with both walls. Thiswill be significantly more complicated combinatorially to analyse. Additionallythere are many experimental results on confined polymers (for example [14, 60,65, 72]). It would be interesting if some comparisons to our results could be made.156Chapter 7Two Directed Walks in the Slitwith Double InteractionsThe final model we study in this manuscript is a natural extension of the single in-teraction model from Chapter 6. It is also a fitting culmination of the work presentin this manuscript as it utilises all the techniques we described previously.In a paper by Alvarez et al. [1], transfer matrix methods was used to investigatethe forces between a polymer and confining walls via a self-avoiding walk model.The qualitative similarities of the results between the models presented here andthe models investigated by Alvarez et al. [1] provides a natural motivation for themodel we study in this chapter. We present a summary of the results from Alvarezet al. [1] and highlight the similarities with our models.Recall that a self-avoiding walk is a sequence of steps in the lattice that nevervisits the same vertex twice (Figure 1.5). The naturally creates volume exclusionproperty, which makes it a realistic choice as a polymer model. (For example,see Chapter 11 of [4]). In a general setting, the complexity of self-avoiding walksmeans they are often resistant to analytical approaches, leading to many open prob-lems with few rigourous results available [73].In a slit of finite width, the self-avoiding walk model is amenable to a transfermatrix approach as the number of possible configurations in each column is finite.Alvarez et al. [1] incorporated single interactions into the model by weighting thewalk by the number of visits to each wall (Figure 7.1).157Figure 7.1: A self-avoiding walk in an interactive slit with single interactionson both walls.At each fixed width, they constructed the transfer matrix for all possible columnconfigurations. They determined numerical estimates of the free energy using thepower method and used the results to determine the force exerted by the polymerson the walls.The force diagram obtained by Alvarez et al. [1] for the self-avoiding walkmodel (Figure 7.2) is remarkably similar to that of the single directed walk in aslit model (Figure 4.5). The similarities in their behaviour can be understood asfollows. In a slit, the expansion of a self-avoiding walk is restricted by two walls.As we increase the length, the walk is forced to expand in the direction of the slit.Hence, the location of the final vertex experiences a net “drift” in one directionas the length increases. The “drift” of the final vertex in a directed walk is builtinto the model. In a broad sense, the similarities between the two models is aresult of the directed walk model capturing the directed “drift” component of theself-avoiding walk model.This suggests directed walks could be used as a physically plausible way tomodel polymer behaviour in a slit. As we have seen in previous chapters, directedwalk models are more amenable to analytic methods and often produce exact solu-tions, whereas self-avoiding walk models do not.The same similarities can be seen when we consider the self-avoiding polygon158Figure 7.2: (Figure 6, [1]) Force diagram of a self-avoiding walk in a slit withsingle interactions.model. Recall that a self-avoiding polygon is a self-avoiding walk that begins andends at the same vertex. Alvarez et al. [1] showed that a self-avoiding polygon in aslit with single interactions only exhibits repulsive forces. This is analogous to ourconclusions for two directed walks in a slit with single interactions (Chapter 6).The same “drift” argument can be applied to justify why two directed walks in aslit might be a physically plausible method of modelling a ring polymer in a slit.The next model considered by Alvarez et al. [1] is a model of ring polymersusing a self-avoiding polygon in a slit with double interactions. The essential dif-ference in this model is that wall interactions occur whenever the walk visits thewall as well as when the walk is a single vertex away from the wall (Figure 7.3).This change produces vastly different behaviour in the force diagram. The numer-ical results for small widths obtained by Alvarez et al. [1] is shown in Figure 7.4.One observable difference between the single and double interaction modelsfor self-avoiding polygons is the existence of an attractive region in the doubleinteraction regime that is not present in the single interaction regime. One inter-pretation of the attractive region is as follows. The additional interactions allowthe entire ring polymer to be adsorbed to a single wall and the attraction poten-159Figure 7.3: A self-avoiding polygon in an interactive slit with double interac-tions on both walls. Wall interactions happen when the walk visits thewall and when the walk is one vertex away.tial is sufficient to overcome entropic repulsion. The second layer of interactionnegates the “shielding” effect observed in the single interaction case for both theself-avoiding polygon model and the two directed walk model.We saw previously that directed walk models can produce qualitatively sim-ilar results to self-avoiding polygon models. This naturally leads us to developa directed walk model with double interactions that is physically consistent withthe self-avoiding polygon model. Further, we hope to use the additional analytictractability of directed walk models to determine exact solutions of the system andprovide a more complete understanding of the behaviour of the system in the largewidth case and in the limiting case.We begin by extending the model for two directed walks in the slit from Chap-ter 6 to incorporate double interactions.7.1 ModelWe consider pairs of directed paths of equal length in a strip of width w and takingsteps (1,1) or (1,−1). These paths may share a vertex but not cross. We considerthose paths with initial vertices at (0,0) and (0,w) (Figure 7.5). Let ϕ be such apair of paths and denote the length |ϕ|. As per the previous model in Section 6.1,160Figure 7.4: (Figure 15, [1]) Force diagram of a self-avoiding polygon in a slitwith double interactions.we add the energies −εa and −εb for each visit of the walks to the bottom and topwalks respectively.Figure 7.5: A pair of Dyck paths in a slit of width 4 and length 18.We expand upon this model by adding energies for double visits to the wall• −εd for each simultaneous visit of both walks to the bottom wall, and• −εe for each simultaneous visit of both walks to the top wall.The number of simultaneous visits of both walks to the bottom wall will bedenoted md(ϕ) while the number of simultaneous visits of both walks to the top161wall will be denoted me(ϕ). The corresponding partition function is defined asZn(w;a,b,d,e) =∑ϕexp[εama(ϕ)kBT· εbmb(ϕ)kBT· εdmd(ϕ)kBT· εeme(ϕ)kBT](7.1.1)=∑ϕama(ϕ) ·bmb(ϕ) ·dmd(ϕ) · eme(ϕ), (7.1.2)where T is the temperature and kB is the Boltzmann constant. The new interactionenergy can be converted into their corresponding Boltzmann weights d = exp[εdkBT]and e = exp[εekBT](Figure 7.6).In its generality, the energies−εd and−εe are independent parameters resultinga four dimensional parameter space. However, there are two specialisations of εdand εe that are of physical interest.The single interaction model in Section 6.1 can be recovered by looking at thespecial case−εd =−εa and−εe =−εb. A second specialisation with−εd =−2εaand −εe =−2εb is also of physical interest. It represents a choice of energies thatis equivalent to an energy of−εa for each visit of each walk to the bottom wall and−εb for each visit of each walk to the top wall (Figure 7.7). Translating this intoits corresponding Boltzmann weights, we getZn(w;a,b) =∑ϕama(ϕ)+2md(ϕ) ·bmb(ϕ)+2me(ϕ). (7.1.3)In addition to reducing the dimension of a parameter space, this particular special-isation reflects a sense of homogeneity between the two walks. That is, each walkinteracts with each wall identically and independently. This mimics the interactionsfor a homogeneous self-avoiding polygon in a slit.Figure 7.6: A pair of Dyck paths in a slit with four independent interactions.162Figure 7.7: A pair of Dyck paths in a slit with double interactions.7.2 Functional EquationsWe attempt to analyse the model using a functional equation approach analogousto that of the previous models. We define a grand canonical partition functionF(r,s;a,b,d,e)≡ F(r,s) = ∑n≥0Zn(w;a,b,d,e)zn (7.2.1)=∑ϕama(ϕ) ·bmb(ϕ) ·dmd(ϕ) · eme(ϕ)z|ϕ|, (7.2.2)where r,s denote the final heights of the walks as per the previous model. We es-tablish the functional equation using a column by column construction. First, weestablish the non-interacting base case, then add in the single interaction compo-nents to obtain the single interaction model (Equation 6.2.15). We then incorporatethe double interaction components by accounting for the updated weights of thewalks under the new interactionsF(r,s) =1+ z(s+ 1s)(r+ 1r)·F(r,s)− zr(s+ 1s)·F(0,s)− zs(r+ 1r)·F(r,0)+ zsr·F(0,0)− zsr · swFd(rs)+ z(b−1)(r+ 1r)[s1]{F(r,s)}+ z(a−1)(s+ 1s)[r1]{F(r,s)}+ z(a−1)(b−1)[s1r1]{F(r,s)}+ z(d−a) · sw[r1sw−1]{F(r,s)}+ z(e−b) · rw[rw−1s1]{F(r,s)} .(7.2.3)163This functional equation is obtained by first looking at the single interactioncase Equation 6.2.15, which accounts for all but the last two terms. The last twoterms are included to account for the new weights, which occur when both walksvisit the same wall. These configurations are reweighted from a to d for the bot-tom wall or from b to e for the top wall. Figure 7.11 shows a schematic of theseconfigurations.We rearrange the equation to obtain the kernel as well as eliminating terms in-volving coefficients of [ris j]F(r,s) where i, j ∈ {1,w−1} using methods describedin Section 4.1 and Section 4.2. This reduces the number of unknown functionsleaves only those involving [ris j]F(r,s) where i, j ∈ {0,w}. Explicitly,K(r,s) ·F(r,s) = 1ab− zrs · swFd(rs)+A(r,s)F(0,s)+B(r,s)F(r,0)+C(r,s)F(0,0)+(1b− 1e)· rw · [rw]F(r,0)+(1a− 1d)· sw · [sw]F(0,s) (7.2.4)whereK(r,s) = 1− z(r+ 1r)(s+ 1s),A(r,s) = a−1a− zr(s+ 1s),B(r,s) = b−1b− zs(r+ 1r),C(r,s) = zrs− b−1ba−1a. (7.2.5)We attempt to use the kernel method on Equation 7.2.4 to find solutions for rˆand sˆ. Since the kernel is the same as the single interaction case (Equation 6.3.2),we apply the same symmetries to generate four equations. Using a similar “orbit164sum” technique, we getK(r,s) · (linear combination of F) =1ab+(rs(a−1)(b−1)abz(r2 +1)(s2 +1) −(a−1)(b−1)ab)·F(0,0)+(d−ad)·[(s2w+2−1)asw(s2−1) −sr(s2w−1)(b−1)abzs2w(r2 +1)(s2−1)]· [sw]F(0,s)+(e−be)·[ (r2w+2−1)brw(r2−1) −sr(r2w−1)(a−1)abzr2w(r2 +1)(s2−1)]· [rw]F(r,0)L(z;r,s,a,b) ·Fd(rs)+M(z;r,s,a,b) ·Fd( sr)+N(z;r,s,a,b) ·Fd (rs)+P(z;r,s,a,b) ·Fd(1rs)(7.2.6)where L(z;r,s,a,b), M(z;r,s,a,b), N(z;r,s,a,b), and P(z;r,s,a,b) are easily com-putable, though complicated functions. As per the previous models, we can makeuse of the kernel to eliminate z by finding solutions rˆ and sˆ such that K(rˆ, sˆ) = 0.After clearing denominators, we get0 = de · sˆ2wrˆ2w(rˆ4−1)(sˆ4−1)−d(e−b) · sˆw(sˆ4−1)(rˆ2 +1)((a−1− rˆ2)rˆ2w−((a−1)rˆ2−1))· [rw]F(r,0)− e(d−a) · rˆw(rˆ4−1)(sˆ2 +1)((b−1− sˆ2)sˆ2w−((b−1)rˆ2−1))· [sw]F(0,s)+de · sˆ2w+2rˆw(b−1− sˆ2)((a−1)rˆ2−1)·Fd(1rˆsˆ)+de · rˆw+2((b−1)sˆ2−1)(a−1− rˆ2)·Fd (rˆsˆ)−de · sˆ2w+2rˆw+2(b−1− sˆ2)(a−1− rˆ2) ·Fd(rˆsˆ)−de · rˆw((b−1)sˆ2−1)((a−1)rˆ2−1)·Fd(sˆrˆ). (7.2.7)The general model does not admit vertical symmetry. Hence, the symmetrya↔ b also requires d↔ e. We derive the analogous result to Equation 6.3.34 usingthe same process. This allows us to eliminate an addition term in Equation 7.2.7.However, we were unable to progress further and obtain simple solutions for rˆ and165sˆ similar to Equation 6.3.22.We attempted to simplify the model in multiple ways to obtain analytical re-sults. We considered:1. The symmetric case where a = b and d = e. Physically, this represents thecase with homogeneous walls with vertical symmetry in the partition func-tion.2. The simplified case where d = a2 and e = b2. This represents a systemwhere the walks interact with each wall independently. This model reducesthe partition function to a two parameter system.3. The combination of the above two cases. This models a system with homo-geneous walls, which the walks interact with independently. This system isreduces the partition function to a function of a single parameter.However, we were unable to produce equations analogous to that of Equa-tion 6.3.22 even with such simplifications. The kernel method requires enoughsymmetries in the system to eliminate the boundary terms of the functional equa-tion. In this instance we were unable to find sufficient symmetries in the system toaccomplish that. The leaves us with an overdetermined system of equations wherewe were unable to extract any meaningful solutions for rˆ and sˆ.Instead, we consider a different approach to analyse the model. FollowingAlvarez et al. [1], we study the model via transfer matrices. The column by columnconstruction we used to establish the functional equations can be easily adapted totransfer matrices. As we shall see, this approach allows us to obtain numericalresults, which is an important step in understanding the behaviour of the system.7.3 Transfer Matrix ApproachA transfer matrix approach allows us to utilise tools from linear algebra to deter-mine a numerical estimate of the dominant eigenvalue for the system at a fixedwidth with rigourous bounds. However, the drawback of this approach is that wewill be unable to explicitly determine the dependence of the dominant eigenvalueon width.166(a) (b)Figure 7.8: (Left) The non-crossing condition. (Right) The parity condition.For each fixed even width w, we define a transfer matrix Aw(a,b,d,e) indexedin the possible final heights of the walks with entries in the interaction parameters.This is analogous to the catalytic variables r,s in the functional equations. Thestates of the transfer matrix is the set of possible final heights of the walks given byS = {(r,s) ∈ Zw+1×Zw+1 | r+ s≤ w, r+ s≡ 0 mod 2} (7.3.1)where r,s represent the possible final heights of the walks. The two conditions arethe non-crossing and parity condition respectively (Figure 7.8). In situations whereno ambiguities arise, we simply denote this Aw for ease of notation. This showsAw is a transfer matrix of size O(w2×w2). Analogous to the functional equationapproach (Equation 7.2.4), we build the transfer matrix Aw in three stages:1. The matrix Bw contains the non-interacting base case (Figure 7.9),2. The matrix Cw(a,b) contains the single interaction components (Figure 7.10),3. The matrix Dw(a,b,d,e) contains the double interaction components (Fig-ure 7.11).For s, t ∈S , we will denote s ∼ t if it is possible to reach append a step to walksending at state s to reach state t = (t1, t2). This gives an explicit construction for167(a) (b) (c) (d)Figure 7.9: The no interaction case has weight w = 1 for all possible transi-tions.the three matrices:Bw[s, t] =1 s∼ t0 otherwise(7.3.2)Cw[s, t] =a−1 s∼ t and t1 = 0, t2 6= 0b−1 s∼ t and t2 = 0, t1 6= 0(a−1)(b−1) s∼ t and t = (0,0)0 otherwise(7.3.3)Dw[s, t] =d−a s∼ t and t = (0,w)e−a s∼ t and t = (w,0)0 otherwise. (7.3.4)These matrices are analogous the functional equation counterparts and are con-structed using the same approach. The final transfer matrix Aw(a,b,d,e) will bethe sum of all the above componentsAw(a,b,d,e) = Bw +Cw(a,b)+Dw(a,b,d,e). (7.3.5)This decomposition is an alternative method for recovering numerical results inthe non-interacting case (Bw) and also in the single interaction case (Bw +Cw(a,b)).The transfer matrix T (z) = zAw since each step contributes z to the overall size168(a) w = a−1 (b) w = b−1 (c)w = (a−1)(b−1)Figure 7.10: Steps in the single interaction case with additional weights w.(a) w = (d −a)(b) w=(e−b)Figure 7.11: Steps in the double interaction case with additional weights w.of the walk. The matrix T (z) is irreducible since every state is a finite number ofsteps away from the state (0,0). However, the matrix is periodic for Dyck paths,with period 2. Hence, we modify our transfer matrix and perform our calculationson the aperiodic matrix T (z) = z2 · [Aw]2.The Transfer Theorem (Lemma 2.22) and Perron-Frobenius (Theorem 2.24)reduces the problem of finding the growth rate of the system to the problem offinding the largest positive eigenvalue of the transfer matrix.This allows us to find the free energy of the system. Ideally, we would like tofind the dominant eigenvalue λ (a,b,d,e) as a function of the interaction param-eters. However, this is a very difficult task and instead, we begin by finding thedominant eigenvalue numerically for specific values in the parameter space. Thisspecification reduces the matrix Aw to a non-negative, real value matrix.Starting with a positive vector, the power method provides increasing accu-169rate dominant eigenvector approximation. At each iteration, a theorem by Collatz[13] (Theorem 2.27) returns a rigourous bound for the precision of the estimation.We terminate this process when Theorem 2.27 returns an interval containing thedominant eigenvalue λ smaller than some predetermined ε > 0. We denote this in-terval Iw(ε;a,b,d,e), or just simply Iw with implicit parameters. That is, we knowλ ∈ Iw(ε;a,b,d,e). The interval provides a maximum uncertainty to the approxi-mation, which will be essential to computing the force in the next section.In principle, this methodology is applicable to sample any point in the fourdimensional parameter space. However, we will specialise to the case described atthe beginning of the model section where each walk interacts independently witheach wall. By specialising to d = a2 and e = b2, we reduce the adjacency matrixto a two dimensional parameter space to produce comparable results to that ofAlvarez et al. [1] and more specifically those depicted in Figure 7.3.7.3.1 Different WidthsUltimately, we are interested in the force diagram of this double interaction system.We can rephrase the definition of force (Definition 4.3) in terms of the dominanteigenvalues of the system to getF (w) = 1λ (w)∂λ (w)∂w . (7.3.6)However, this requires knowing the dependence of λ on the width. Since we canobtain information about the system at integer widths, we considered the discretederivative form of the above equation. By denoting λw to be the dominant eigen-value of the transfer matrix at width w, we instead defineF (w) = 1λwλw+2−λw2.. (7.3.7)At a fixed width, the sign of the force for any given set of parameters is determinedby the difference between λw+2 and λw at those parameters. This requires thecomparison of the model at two different widths. In particular, the set of parametervalues a,b where λw = λw+2 are of particular interest. The set of such values forma zero force curve in the parameter space where F (w) = 0. That is, the system170walls experience no force exerted by the walls at these values. This partitionsthe parameter space into regions where the walks exert an expansive force on thewalls (F > 0) and contractive force on the walls (F < 0). Further, determiningthe shape of this curve allows a direct qualitative comparison with the analogousresults shown in Figure 7.3.We searched for the set of zero force points using a bisection method on onedimensional slices of the parameter space. More specifically, we took one dimen-sional slices the form b(a) = a+k for a fixed constant k and looked for an intersec-tion with the zero force curve. This reduces the problem of determining the zeroforce curve over a two dimensional space to finding the intersection point betweenthe zero force curve and a linear function in a single parameter (Figure 7.12).Figure 7.12: The value a = a0(w,k) where a slice b = a+k intersects the zeroforce curve.This motivates the following definition.Definition 7.1. For a given width w and a fixed constant k ∈ R, the value a =a0(w,k) defines a zero force point given by (a,a+k) in the parameter space whereF (w) = 0.171In principle, there is no guarantee of the existence or uniqueness of zero forcepoints for arbitrary choice of w,k. However, based on the observations from theself-avoiding walk models, we anticipate that the zero force curve intersects uniquelywith slice of this form. In what follows, we will develop the necessary machineryto determine a numerical estimate of a0(w,k), provide a provable uncertainty to theestimate, as well as provide evidence for the existence and uniqueness of the pointa0(w,k).For each width w and choice of k, we can obtain an estimate of the dominanteigenvalue λw(a) and λw+2(a) for any choice of parameter value a. Further, byspecifying an uncertainty ε > 0, we determine rigourously, intervals Iw(ε;a) andIw+2(ε;a) which contains the dominant eigenvalue in the respective widths. Thismotivates the following notation. Given a fixed width w,k, and parameter value a,we define the followingIw ≺ Iw+2 If ∀µ ∈ Iw and ∀ν ∈ Iw+2 we have µ < ν ,Iw ≈ Iw+2 If Iw∩ Iw+2 6= /0,Iw  Iw+2 If ∀µ ∈ Iw and ∀ν ∈ Iw+2 we have µ > ν .For each value a such that Iw(a)≈ Iw+2(a), we must have |λw+2(a)−λw(a)|<2ε . That is, the point (a,b(a)) is a candidate for the zero force point in the param-eter space.The dominant eigenvalue λw(a) and λw+2(a) are continuous functions in a byLemma 4.8. Hence, the difference λw+2(a)−λw(a) is also continuous. Supposewe can find two values a1,a2 such that Iw(a1) ≺ Iw+2(a1) and Iw(a2)  Iw+2(a2).Then the intermediate value theorem states that there must exist a value ac suchthat Iw(ac)≈ Iw+2(ac). Further, there exists a range of values (amin,amax) such thatfor any ac ∈ (amin,amax), we have Iw(ac)≈ Iw+2(ac) (Figure 7.13).Starting with two values a1,a2 with Iw(a1) ≺ Iw+2(a1) and Iw(a2)  Iw+2(a2),we use the bisection method to find a value ac between a1 and a2 such that Iw(ac)≈Iw+2(ac). The value ac is an estimate of the zero force points along our slice. Thatis|ac−a0(w,k)| ≤ |amax−amin| . (7.3.8)172Figure 7.13: The interval R is the interval of a where Iw(a) and Iw+2(a) havea non-empty intersection.This leaves us with the task of finding estimate on amin and amax. We ac-complish this by repeating a similar bisection on the interval (a1,ac) and (ac,a2)respectively until the intervals are each smaller than some predetermined δ . Byaccounting for the potential boundary cases, we obtain an interval R = (amin−δ ,amax +δ ) that contains a0(w,k). We have for any choice of ac,|ac−a0(w,k)| ≤ |R|. (7.3.9)We use the value ac as our approximation for a0(w,k).1737.4 Results and DiscussionIn this section, we present the numerical results we obtained using the methoddescribed in the previous section. For the Dyck path model, we usek ∈0, ±0.05, ±0.1, ±0.2,±0.25, ±0.3, ±0.4, ±0.5,±0.7, ±0.8, ±0.9, ±1.0,±1.2, ±1.5, ±2.0, ±3.0.Figure 7.14: A visual representation of the non-negative slices taken (k≥ 0).A visual representation of the different slices (k values) taken is shown in Fig-ure 7.14. At a fixed width w and constant k, we compute a rigourous interval Rcontaining the value a0(w,k) using the method described in the Section 7.3. Basedon the results from previous models, we expect this model to exhibit short rangeattraction/repulsion effects. That is, we anticipate the dominant eigenvalue to beexponentially dependent on the width. As width increases, we expect the differ-ence λw+2(a)−λw(a) to decay exponentially. As a result, we require increasinglyprecise estimates of the dominant eigenvalue. We targeted a resulting interval R174of machine precision |R| = O(10−16). To accomplish that, we required the useof multiprecision integer package MPIR [25] to compute the dominant eigenvalueprecise to roughly ε = O(10−100). We provide a better picture of the zero forcecurve by interpolating between the known points. An example is shown in Fig-ure 7.15. The size of the interval |R| is insignificant relative to the size of the pointa0(w,k) plotted on the graph and hence we omit error bars from our plots.At this point, we have enough machinery to provide some justification for Def-inition 7.1. In particular, we believe the value a0(w,k) exists and is unique for eachchoice of w,k. We first provide evidence for its existence as follows. For each fixedwidth w and |k| ≤ 3, we found that λw+2(a)− λw(a) < 0 for small a values (I.ea≈ 1) and λw+2(a)−λw(a)> 0 for large values of a (I.e. a > 10). The continuityof λw+2(a) and λw(a) ensures that we have at least one value a0(w,k), indicatingthat the zero force curve intersects the line b(a) = a+ k at least once.Next, we tried looking values w′, k′ such that at width w′, the line b(a) =a+ k′ intersects the zero force curve multiple times. That is, we attempted to findvalues of w′ and k′ that resulted in multiple values of a0(w′,k′). We divided theinterval a into twenty subintervals and looked at the sign of λw+2(a)−λw(a) at theboundaries of the subintervals. A change in sign across an interval would indicatethe existence of a0(w′,k′) in that interval. However, for each choice of w and k, weonly found a single subinterval where this behaviour occurred. Further, numericalevidence suggests that λw+2(a)−λw(a) and in increasing function in a. This leadsus to believe that the value a0(w,k) is unique for a fixed w, k.We repeated the procedure for systems of higher widths. The results are shownin Figure 7.16 and Figure 7.17. The figure shows a “spike” that grows along theslice a = b. It remains an open problem to determine the behaviour of the zeroforce point along this slice as w → ∞. Equivalently, the existence of the limitlimw→∞a0(w,0) is currently unknown.However, Figure 7.16 and Figure 7.17 appear qualitatively similar to Figure 7.3.Based on these observations, we predict the limiting zero force curve (as w→ ∞)will fall into one of the four categories shown in Figure 7.18 depending on thebehaviour of a0(w,k) for k = 0 (along the line a = b) and for small values of k.This naturally leads to the problem of looking at the behaviour of the zero forcecurve around the line a = b. In particular, for what values of k does limw→∞a0(w,k)175Figure 7.15: (left) A plot of the zero force point estimates for w = 4 in thea− b plane. (right) The interpolation of the known points to form azero force curve estimate.exist. For this problem, we looked at the behaviour of a0(w,k) for a fixed value ofk and then increasing the width w.Figure 7.19 and Figure 7.20 shows the value of a0(w,k) for different values ofk as width increases. It appears that for k ≥ 0.2, the values a0(w,k) appear to con-verge towards a finite limit. However, from this plot alone, it is not clear whetherthe a0(w,0.1) converges. We explore this further by looking at the system at higherwidths. Figure 7.21 shows the results for k values close to 0 and Figure 7.22 showsthe values of a0(w,0).From Figure 7.21, we notice for two different values k1 and k2, the difference|a0(w,k1)−a0(w,k2)| increases as the width w increases. This suggests the infiniteslit behaviour is more “spike” shaped than “bubble” shaped.The limiting behaviour of a0(w,0) remains unclear from Figure 7.22. That is,whether the zero force point along a = b converges or diverges as width increases.If the value a0(w,0) diverges, the resulting behaviour of the force diagram will looksimilar to Figure 7.18(d).One method we have for determining convergence of a sequence is to considerthe first differences of a0(w,k) as a function of width. That is, if the differences∆a0 = a0(w+2,k)−a0(w,k) do not approach zero, then a0(w,k) cannot be a con-176Figure 7.16: Plots of the zero force curve estimates for small widths.vergent sequence. To assist our fit, we consider a plot against inverse width(w−1).This gives us an estimate of the limit as w tends to infinity by extrapolating the datato the y-intercept.The line of best fit from Figure 7.23 is obtained by omitting the data pointsfor width (w < 40) to remove any irregularities caused by small widths. The lineappears to be consistent with the data observed at large widths. The data appearsto have a non-zero intercept, which implies divergent behaviour. Further, thereappears to be a linear relation between the first differences and inverse of the width(w−1). This behaviour suggests a logarithmic dependence of a0(w,0) on the width.Hence, we plotted the values a0(w,0) against logarithmic width (log(w)) to verifyour hypothesis (Figure 7.24).Figure 7.24 shows the line of best fit for a0(w,0) against the logarithm of width.For the line of best fit, we omitted the zero force points for small widths (w < 40).The line appears to be consistent with the values a0(w,0) obtained at large widths.This suggests the values a0(w,0) has a dependence on log(w). Further, Figure 7.24shows a slight upwards curvature in a0(w,0) as width increases. This indicates177Figure 7.17: Plots of the zero force curve for larger widths.the exsistence of a small super-logarithmic component in the behaviour in the zeroforce points in the infinite slit limit. The non-zero intercept predicted based onFigure 7.23 suggest that this small super-logarithmic component could be linear.While we cannot determine the zero force curve analytically, we have shownstrong evidence that a0(w,0) diverges in the limit w→ ∞. At the same time, thepoints a0(w,k) appears to converge for all k 6= 0. Further, for values k1 6= k2, thedifference |a0(w,k1)−a0(w,k2)| appears to be increasing, which suggests that zeroforce curve for this model is closer to a “spike” shape than the “bubble” shapeobserved for the self-avoiding polygons.In Chapter 4, we saw that the zero force curve of the single walk model satisfiedthe simple equation ab− a− b = 0. We would like to find an analogously “nice”equation for the zero force curve of this model. That is, we would like to finda polynomial in parameters a and b, of low degree with small coefficients that issatisfied by the zero force curve. However, we have no a priori reason to believesuch an equation exists nor what its form might be. Hence, we attempted to makepredictions to what such an equation might be based on the numerical data we have178(a) A limiting bubble. (b) A limiting spike.(c) An infinite bubble, zero force curvehas asymptote away from a = b.(d) An infinite spike, zero force curveasymptotically approaching a = b.Figure 7.18: Possible limiting zero force curves as w→ ∞.collected thus far.Consider the single walk model (Chapter 4). The zero force curve intersectsthe line b = 2 at a = 2. For the two walk model (Chapter 6), we found no zeroforce curve. However the point (a,b) = (2,2) in the parameter space producedan exact solution for the force exerted on the wall. Following this idea, if the zeroforce curve is satisfied a simple polynomial, then the solution to this equation whenb = 2 should yield an algebraic solution for a. In other words, the solutions for aafter we substitute b = 2 into the zero force curve will give us information aboutthe nature of the curve.179Figure 7.19: Location of the zero force points as width increases for large kvalues where they appear to converge.Figure 7.20: Location of the zero force points as width increases for smallerk values where not every point appears to converge.To investigate this, we are deviate from the diagonal slices and look at alongthe horizontal slice of b = 2 for values ac(w) where the force F (w) = 0 (SeeFigure 7.25). We looked at the intersection of the zero force curve along this linefor systems at different widths. Figure 7.26 shows the values of ac(w) as widthincreases. In this case, the values ac(w) along this slice appears to converge tothe value 3.3009887454668760 . . .. Unfortunately, we were unable to find any180Figure 7.21: Values a0(w,k) for different k values close to 0 as width in-creases.Figure 7.22: Values a0(w,0) as width increases where the points do not ap-pear to converge.meaningful interpretation of this number via an inverse symbolic calculator [56].However, this value is remarkably close to a value quoted in Section 5.6.2[55]. Recall for the model of two walks above an interacting wall, we observetwo phase transitions in a fixed energy ratio model (r-model) That is d = ar. Inparticular, we observed that for r = 2 (d = a2), the first transition occurs at a = 2,where the system goes from a desorbed phase to a-rich phase. The second phasetransition occurs at a = 3.301, where the system goes from the a-rich phase to181Figure 7.23: Plot of first differences ∆a0 against inverse width with the lineof best fit.Figure 7.24: Plot of a0(w,0) against Logarithm of the width (log(w)) withthe line of best fit.the d-rich phase. This quoted value is obtained solving the complicated equationEquation 5.5.20 numerically and does not appear to be the solution of any “nice”equation.At first glance, these numbers appear to be the result of two different processes.Our numerical result predicts a particular specification of the slit model where thewalks do not exert any force on the walls. On the other hand, the result based on afixed energy ratio model predicts a second phase transition in the half plane modelat a specific value of a. The discussion in Chapter 6 shows that it is possible fora two walks in the slit model to exhibit behaviour that mimics two noninteracting182Figure 7.25: A schematic showing the intersection of the zero force curvewith the line b = 2.single walk models (For example, See Equation 6.5.3). While we cannot provethat these two numbers are identical, we believe that the proximity of these valuesis unlikely to be coincidental and it indicates the existence of a deeper connectionbetween the two phenomena.If there exists a polynomial of low degree and small coefficients that is satisfiedby the zero force curve, then the solution of a for curve when b = 2 should producean algebraic number. However, the convergence towards 3.3009887454668760as w→ ∞ suggests the zero force curve does not intersect the line b = 2 at analgebraic number. Hence, this indicates that the zero force curve is not satisfied bya polynomial of low degree with small coefficients.7.5 ConclusionsWe extend a previous directed walk model of two polymers confined to a longmacroscopic sized slit with sticky walls to allow interaction between both walkswith both walls. We were unable to produce a meaningful analysis of the modelusing a kernel method approach analogous to the case for single interactions. How-183Figure 7.26: Convergence of the zero force point estimate along the line b= 2towards a = 3.30098 . . ..ever, we were able to approximate the shape of the zero force curve with rigourousuncertainties using a numerical transfer matrix approach. We observed the exis-tence of both attractive and repulsive regimes in this model.We have shown strong evidence that the points of zero force along the linea = b diverges in infinite slit limit. Meanwhile, the points of zero force along theline b(a) = a + k appears to converge for all k 6= 0. Further, the data obtainedsuggests that zero force curve for this model is closer to a “spike” shape than the“bubble” shape observed for the self-avoiding polygons.In the limit as w→ ∞, the intersection of the zero force curve with the lineb = 2 approaches 3.3009887454668760, a value that is remarkably close to a valueobtained in Section 5.6.1. It is likely that the two values are indeed the same,indicating a deeper connection between the two models. 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Substituting this into theexpression and manipulating gives(xeit)v+2− 1(xeit)v+2(xeit)v− 1(xeit)v= (xv+2− x−(v+2))cos((v+2)t)+ i(xv+2 + x−(v+2))sin((v+2)t)(xv− x−v)cos(vt)+ i(xv + x−v)sin(vt) .(A.0.1)By multiplying the denominator by its complex conjugate we obtain an expressionof the form (P(x)+ iQ(x))/D(x) andP(x) = (x2v+2 + x−(2v+2))cos(2t)− (x2 + x−2)cos((2v+2)t) (A.0.2)Q(x) = (x2v+2− x−(2v+2))sin(2t)− (x2− x−2)(sin((2v+2)t) (A.0.3)D(x) = (x2v + x−2v)−2cos(2vt). (A.0.4)Note that P,Q,D are all real. Hence this expression is real if and only if Q(x) = 0.It is clear that if rˆ ∈ R (t = 0,pi) or if rˆ is a complex number of unit magnitude192(x = 1), then Q(x) = 0. Thus, suppose that there is a value rˆ that does not satisfyeither case (t 6= 0,pi and x 6= 1), then0 = Q(x) = (x2v+2− x−(2v+2))sin(2t)− (x2− x−2)(sin((2v+2)t), (A.0.5)which givesx2v+2− x−(2v+2)x2− x−2 =sin((2v+2)t)sin(2t) . (A.0.6)The left hand side can be expanded to give the sumx2v+2− x−(2v+2)x2− x−2 = x−2vv∑i=0x4i (A.0.7)with v+1 summands. When v is even, the sum expands tox2v + x2v−4 + . . .+ x4 +1+ x−4 + . . .+ x−2v+4 + x−2v (A.0.8)and in the case where v is odd, the sum expands tox2v + x2v−4 + . . .+ x6 + x2 + x−2 + x−6 + . . .+ x−2v+4 + x−2v. (A.0.9)In each case, the summands can be pairs off in the form x2l +x−2l for the appropri-ate values of l and a remaining 1 when v is even. Now, for for x 6= 1 and a positiveinteger k, we have xk + x−k > 2. Summing over all pairs, we getx2v+2− x−(2v+2)x2− x−2 > v+1. (A.0.10)For the right hand side, we havesin((2v+2)t)sin(2t) =ei(2v+2)t − e−i(2v+2)tei2t − e−i2t , (A.0.11)and by substituting q = e2it we getsin((2v+2)t)sin(2t) =qv+1−q−(v+1)q−q−1 . (A.0.12)193When expanded, this givesqv+1−q−(v+1)q−q−1 = q−vv∑i=0q2i. (A.0.13)Similar to the case with x, When v is even, this sum expands toqv +qv−2 + . . .+q2 +1+q−2 + . . .+q−v+2 +q−v (A.0.14)and in the case where v is odd, the sum expands toqv +qv−2 + . . .+q3 +q+q−1 +q−3 + . . .+q−v+2 +q−v. (A.0.15)In either case, the powers of q can be paired up and simplified as followsql +q−l = e2ilt + e−2ilt = 2cos(2lt). (A.0.16)Thussin((2v+2)t)sin(2t) =1+2∑v/2j=1 cos(2 jt) v even2∑(v−1)/2j=1 cos(2(2 j+1)t) v odd.(A.0.17)In either case, each summand can be bounded above by 1 and given the number ofsummands, we can conclude thatx2v+2− x−(2v+2)x2− x−2 > v+1≥sin((2v+2)t)sin(2t) . (A.0.18)Thus contradicting the existence of the point rˆ = xeit with x 6= 1 and t 6= 0,pi .194

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