WIAS Preprint No. 2910, (2022)

Gibbsianness of locally thinned random fields


Authors

  • Engler, Nils
  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Külske, Christof

2020 Mathematics Subject Classification

  • 60D05 60K35 82B20

Keywords

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

DOI

10.20347/WIAS.PREPRINT.2910

Abstract

We consider the locally thinned Bernoulli field on ℤ d, which is the lattice version of the Type-I Matérn 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 [46], and cluster expansions

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