Exact solution of some quarter plane walks with interacting boundaries

TL;DR

This paper explores the integrability of quarter plane walks with boundary interactions, providing exact solutions for certain models and highlighting challenges in solving more complex cases.

Contribution

It introduces boundary interactions into quarter plane walk models and derives exact solutions for some cases, expanding understanding of their algebraic and differential properties.

Findings

Exact solutions obtained for some boundary-interacting models

Kernel method limitations identified when counting boundary sites

Generalized Kreweras model solved with boundary counts

Abstract

The set of random walks with different step sets (of short steps) in the quarter plane has provided a rich set of models that have profoundly different integrability properties. In particular, 23 of the 79 effectively different models can be shown to have generating functions that are algebraic or differentiably finite. Here we investigate how this integrability may change in those 23 models where in addition to length one also counts the number of sites of the walk touching either the horizontal and/or vertical boundaries of the quarter plane. This is equivalent to introducing interactions with those boundaries in a statistical mechanical context. We are able to solve for the generating function in a number of cases. For example, when counting the total number of boundary sites without differentiating whether they are horizontal or vertical, we can solve the generating function of a generalised Kreweras model. However, in many instances we are not able to solve as the kernel methodology seems to break down when including counts with the boundaries.