Supervisor: Professor Thomas Prellberg
Project description:
Simplified models of polymers have been studied in the past using Dyck and Motzkin paths and variants. Over the last fifteen years there has been a significant development of new techniques based on functional equations from algebraic combinatorics. The so-called kernel method can be used to solve linear combinatorial functional equations in so-called catalytic variables, allowing to go beyond directed models.
As an example, the exact solution of a lattice model of partially directed walks in a wedge has only been possible using an iterative version of this kernel method, developed the project supervisor. Some very deep insight into certain phase transitions of polymer lattice models can be gained by considering area-weighted lattice models of vesicles, as there is a connection between the scaling behaviour of certain vesicle models and asymptotics of basic (q-deformed) hypergeometric functions and that by analysing the uniform asymptotics of these special functions via contour-integral representations, one can gain an explicit analysis of the phase transition down to the level of scaling functions. A previous PhD project managed to generalise the asymptotic analysis beyond two coalescent saddle points, leading to a hierarchy of scaling functions for the inflation transition of two-dimensional vesicles, confirming a scenario suggested by John Cardy, and opening up a pathway for further developments.
The aim of this proposal is to extend this study to the analysis of other models that are still amenable to similar techniques but that can lie in different universality classes, e.g. by relaxing the planarity restriction. One example is the ``Fighting Fish'' model, which is a combinatorial non-planar model of a random branching surface. Its name is inspired by the Siamese fighting fish betta splendens, which has a highly developed fringe tail.
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