WIAS Preprint No. 2878, (2021)

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


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

2020 Mathematics Subject Classification

  • 60D05 60K35 82B20


  • Gibbsianness, Gibbs-uniqueness, Bernoulli field, local thinning, two-layer representation, Dobrushin uniqueness, Peierls' argument




We 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.

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