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Enumeration problems in directed walk models Wong, Thomas


Self-avoiding walks appear ubiquitously in the study of linear polymers as it naturally captures their volume exclusion property. However, self-avoiding walks are very difficult to analyse with few rigourous results available. In 2008, Alvarez et al. determined numerical results for the forces induced by a self-avoiding walk in an interactive slit. These results resembled the exact results for a directed model in the same setting by Brak et al., suggesting the physical consistency of directed walks as polymer models. In the directed walk model, three phases were identified in the infinite slit limit as well as the regions of attractive and repulsive forces induced by the polymer on the walls. Via the kernel method, we extend the model to include two directed walks as a way to find exact enumerative results for studying the behaviour of ring polymers near an interactive wall, or walls. We first consider a ring polymer near an interactive surface via two friendly walks that begin and end together along a single wall. We find an exact solution and provide 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 walks are confined in an interactive slit. We find and analyse the exact solution of two friendly walks tethered to different walls where single interactions are permitted. That is, each walk interacts with the wall it is tethered to. This model exhibits repulsive force only in the parameter space. While these results differ from the single polymer models, they are consistent with Alvarez et al. Finally, we consider the model with double interactions, where each walk interacts 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 of attractive and repulsive forces. The results we obtain are qualitatively similar to those presented in Alvarez et al. Furthermore, we provide evidence that the zero force curve does not satisfy any simple polynomial equation.

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