Transition of the simple random walk on the graph of the ice-model

TL;DR

This paper analyzes the transition probabilities of a simple random walk on multigraphs derived from the 6-vertex model, revealing an unexpected use of continued fractions in the computation.

Contribution

It introduces a novel method using continued fractions to compute transition probabilities for the simple random walk on 6-vertex model graphs.

Findings

Transition probabilities are explicitly computed.

Continued fractions are effectively used in the analysis.

Results connect combinatorial models with analytical techniques.

Abstract

The 6-vertex model is a seminal model for many domains in Mathematics and Physics. The sets of configurations of the 6-vertex model can be described as the sets of paths in multigraphs. In this article the transition probability of the simple random walk on the multigraphs is computed. The unexpected point of the results is the use of continuous fractions to compute the transition probability.