Mixing Properties for Hom-Shifts and the Distance between Walks on Associated Graphs

TL;DR

This paper investigates the conditions under which the graph of bi-infinite walks on a finite connected graph has finite diameter, relating to the mixing properties of associated hom-shift spaces.

Contribution

It characterizes when the diameter of the walk graph is finite, advancing understanding of mixing properties in hom-shift spaces.

Findings

Identifies conditions for finite diameter of walk graphs

Links graph properties to mixing behavior of hom-shifts

Provides new criteria for analyzing walk distances

Abstract

Let $\mathcal H$ be a finite connected undirected graph and $\mathcal H_{walk}$ be the graph of bi-infinite walks on $\mathcal H$; two such walks $\{x_i\}_{i\in \mathbb Z}$ and $\{y_i\}_{i \in \mathbb Z}$ are said to be adjacent if $x_i$ is adjacent to $y_i$ for all $i \in \mathbb Z$. We consider the question: Given a graph $\mathcal H$ when is the diameter (with respect to the graph metric) of $\mathcal H_{walk}$ finite? Such questions arise while studying mixing properties of hom-shifts (shift spaces which arise as the space of graph homomorphisms from the Cayley graph of $\mathbb Z^d$ with respect to the standard generators to $\mathcal H$) and are the subject of this paper.