Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites

TL;DR

This paper investigates how removing isolated sites from Bernoulli lattice fields affects their Gibbsian properties, revealing a phase transition where high-density fields become non-Gibbsian while low-density fields remain Gibbsian.

Contribution

It demonstrates that a simple thinning operation can induce non-Gibbsianness in Bernoulli fields at high densities, identifying a critical density threshold.

Findings

High-density Bernoulli fields become non-Gibbsian after thinning.

Low-density Bernoulli fields preserve the Gibbs property.

The transition depends on the occupation density p.

Abstract

We consider the i.i.d. Bernoulli field $\mu_p$ on $\mathbb{Z}^d$ with occupation density $p\in [0,1]$. To each realization of the set of occupied sites we apply a thinning map that removes all occupied sites that are isolated in graph distance. We show that, while this map seems non-invasive for large $p$, as it changes only a small fraction $p(1-p)^{2d}$ of sites, there is $p(d)<1$ such that for all $p\in(p(d),1)$ the resulting measure is a non-Gibbsian measure, i.e., it does not possess a continuous version of its finite-volume conditional probabilities. On the other hand, for small $p$, the Gibbs property is preserved.