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- ItemGibbsianness of locally thinned random fields(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Engler, Nils; Jahnel, Benedikt; Külske, ChristofWe 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
- ItemGibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Jahnel, Benedikt; Külske, ChristofWe consider the i.i.d. Bernoulli field μ p on Z d with occupation density p ∈ [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 ∈ (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.