Spatial Mixing for Independent Sets in Poisson Random Trees

TL;DR

This paper investigates how correlation decay and spatial mixing properties in the hard-core model behave on Poisson random trees, revealing thresholds and differences from regular trees.

Contribution

It provides new insights into the spatial mixing thresholds on Poisson trees, showing similarities and differences with regular trees and analyzing the effects of tree finiteness.

Findings

Weak spatial mixing occurs for large $d$ when $\lambda < f(d)$

Strong spatial mixing does not always coincide with weak on Poisson trees

WSM holds almost surely for $d<1.434$ but fails with positive probability for $d>1$

Abstract

We consider correlation decay in the hard-core model with fugacity $\lambda$ on a rooted tree $T$ in which the arity of each vertex is independently Poisson distributed with mean $d$. Specifically, we investigate the question of which parameter settings $(d, \lambda)$ result in strong spatial mixing, weak spatial mixing, or neither. (In our context, weak spatial mixing is equivalent to Gibbs uniqueness.) For finite fugacity, a zero-one law implies that these spatial mixing properties hold either almost surely or almost never, once we have conditioned on whether $T$ is finite or infinite. We provide a partial answer to this question, which implies in particular that 1. As $d \to \infty$, weak spatial mixing on the Poisson tree occurs whenever $\lambda < f(d) - o(1)$ but not when $\lambda$ is slightly above $f(d)$, where $f(d)$ is the threshold for WSM (and SSM) on the $d$-regular tree. This suggests that, in most cases, Poisson trees have similar spatial mixing behavior to regular trees. 2. When $1 < d \le 1.179$, there is weak spatial mixing on the Poisson($d$) tree for all values of $\lambda$. However, strong spatial mixing does not hold for sufficiently large $\lambda$. This is in contrast to regular trees, for which strong spatial mixing and weak spatial mixing always coincide. For infinite fugacity SSM holds only when the tree is finite, and hence almost surely fails on the Poisson($d$) tree when $d>1$. We show that WSM almost surely holds on the Poisson($d$) tree for $d < \mathbf{e}^{1/\sqrt{2}}/\sqrt{2} =1.434...$, but that it fails with positive probability if $d>\mathbf{e}$.