Gibbsianness of locally thinned random fields

TL;DR

This paper proves that a locally thinned Bernoulli lattice field, akin to a hardcore process, has a Gibbsian structure with controlled dependence, applicable across different occupation probabilities.

Contribution

It establishes Gibbsian representation and quasilocal dependence control for the thinned Bernoulli field on b^d, using advanced probabilistic methods.

Findings

Gibbsian representation of the thinned measure.

Control of quasilocal dependence for small and large p.

Application of Dobrushin criteria and cluster expansions.

Abstract

We consider the locally thinned Bernoulli field on $\mathbb Z^d$, which is the lattice version of the Type-I Mat\'ern hardcore process in Euclidean space. It is given as the lattice field of occupation variables, obtained as image of an i.i.d. Bernoulli lattice field with occupation probability $p$, under the map which removes all particles with neighbors, while keeping the isolated particles. We prove that the thinned measure has a Gibbsian representation and provide control on its quasilocal dependence, both in the regime of small $p$, but also in the regime of large $p$, where the thinning transformation changes the Bernoulli measure drastically. Our methods rely on Dobrushin uniqueness criteria, disagreement percolation arguments, and cluster expansions.