Gibbsianness of Locally Thinned Random Fields

N. Engler, B. Jahnel, C. Kuelske

2022, v.28, Issue 2, 185-214

ABSTRACT

We consider the locally thinned Bernoulli field on $\zd$, 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~\cite{BeMa94}, and cluster expansions

Keywords: Gibbsianness, Bernoulli field, local thinning, two-layer representation, Dobrushin uniqueness, cluster expansion, disagreement percolation

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