Strong spatial mixing in homomorphism spaces

TL;DR

This paper investigates conditions under which Gibbs measures on graph homomorphisms exhibit strong spatial mixing, extending previous results by linking dismantlability of target graphs to mixing properties.

Contribution

It provides new necessary and sufficient conditions for strong spatial mixing in homomorphism spaces, strengthening prior results on uniqueness and mixing.

Findings

Dismantlability of the target graph relates to existence of Gibbs measures with strong spatial mixing.

Strengthens previous work by showing unique Gibbs measure can have weak spatial mixing.

Existence of dismantlable graphs where no Gibbs measure exhibits strong spatial mixing.

Abstract

Given a countable graph $\mathcal{G}$ and a finite graph $\mathrm{H}$, we consider $\mathrm{Hom}(\mathcal{G},\mathrm{H})$ the set of graph homomorphisms from $\mathcal{G}$ to $\mathrm{H}$ and we study Gibbs measures supported on $\mathrm{Hom}(\mathcal{G},\mathrm{H})$ . We develop some sufficient and other necessary conditions on $\mathrm{Hom}(\mathcal{G},\mathrm{H})$ for the existence of Gibbs specifications satisfying strong spatial mixing (with exponential decay rate). We relate this with previous work of Brightwell and Winkler, who showed that a graph $\mathrm{H}$ has a combinatorial property called dismantlability if and only if for every $\mathcal{G}$ of bounded degree, there exists a Gibbs specification with unique Gibbs measure. We strengthen their result by showing that this unique Gibbs measure can be chosen to have weak spatial mixing, but we also show that there exist dismantlable graphs for which no Gibbs measure has strong spatial mixing.