Uniform spanning forest on the integer lattice with drift in one coordinate.
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Martínez Dibene, Guillermo Elías
issued: 2020-08-10T14:14:45Z
issued: 2020
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: The purpose of this thesis is to investigate the Uniform Spanning Forest (USF) in the nearestneighbour integer lattice Z^{d+1} = ZxZ^d with the family of conductances c((n, x), (n', x')) = e^{λ max(n,n')}, for all (n, x) ~ (n', x'), where ~ stands here for adjacency of vertices and λ > 0 is a fixed parameter. The random walk corresponding to this assignment of conductances resembles a discrete version of Brownian motion with drift. These conductances are not bounded, neither from below nor from above. This entails that many results known in the literature are not applicable. Our results include: 1. Estimate, up to multiplicative constants, the Green's function of this network. 2. Both the Wired and Free uniform spanning forest measures coincide for these assignment of conductances via a coupling argument. Call USF the resulting measure. 3. For d = 1; 2 we will show that USF consists of a tree; for d ≥ 3; it consists of infinitely many infinite trees. 4. For every d; every component of USF is one-ended. 5. For d ≥ 3; there exists a metric η on Z^{d+1} such that the probability that z, z' belong to the same USF-component is bounded above and below by multiplicative constants of η(z - z')^{-(d-2)}: The upper bound is then generalised to the probability that all the vertices in a finite set should belong to the same component. 6. Collapse every tree in USF to a point and denote by D(z, z') the number of edges that separate the USF-components at z and z': Almost surely, D(z, z') = U+2308 (d-2)/4 U+2309
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/75406
note: Uniform spanning forest on the integer latticewith drift in one coordinate.byGuillermo El´ıas Mart´ınez-Dibene.M.Sc., Mathematics, Centro de Investigacio´n en Matema´ticas, 2013.B.Sc., Mathematics, Universidad de Guanajuato, 2008.A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies.(Mathematics.)THE UNIVERSITY OF BRITISH COLUMBIA.(Vancouver.)July 2020c©Guillermo El´ıas Mart´ınez-Dibene, 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Uniform spanning forest on the integer lattice with drift in one coordinate.submitted by Guillermo Mart´ınez-Dibene in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Mathematics.Examining Committee:Martin Thomas Barlow, MathematicsSupervisorGordon Slade, MathematicsSupervisory Committee MemberHarry Joe, StatisticsUniversity ExaminerAndrew Rechnitzer, MathematicsUniversity ExaminerAdditional Supervisory Committee Members:Ed Perkins, MathematicsSupervisory Committee MemberiiAbstractThe purpose of this thesis is to investigate the Uniform Spanning Forest (USF) in the nearest-neighbour integer lattice Zd+1 = Z× Zd with the family of conductancesc((n, x), (n′, x′)) = eλmax(n,n′), for all (n, x) ∼ (n′, x′),where ∼ stands here for “adjacency” of vertices and λ > 0 is a fixed parameter. Therandom walk corresponding to this assignment of conductances resembles a discrete versionof Brownian motion with drift. These conductances are not bounded, neither from belownor from above. This entails that many results known in the literature are not applicable.Our results include:• Estimate, up to multiplicative constants, the Green’s function of this network.• Both the Wired and Free uniform spanning forest measures coincide for these assign-ment of conductances via a coupling argument. Call USF the result…
has type: Thesis/Dissertation
date issued: 2020-11
is shown at: 10.14288/1.0392676
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
type: Text
identifier URI: http://hdl.handle.net/2429/75406