Transition of the simple random walk on the ice model graph (Q6633185)

Describing a simple random walk on a graph \(G_K\) in the stationary regime involves stating that the distribution of the pair \((Y_K(0),Y_K(1))\) is uniform on the edges \(E_K\). In this paper, the authors provide a formula for the transition probability of the simple random walk starting from a given vertex. Interestingly they find the use of continued fractions to measure the length of the constancy blocks of the digits of the given vertex. A primary consequence of their main result is that the transition probability formula significantly simplifies and accelerates simulations of the walk Then astonishingly their main result also accounts for the case when \(k=\infty\). This clearly gives insight into the decay of the variance of the height function in \(G_\infty\). Then they also derive nicely several interesting results.