November 2023 Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites
Benedikt Jahnel, Christof Külske
Author Affiliations +
Bernoulli 29(4): 3013-3032 (November 2023). DOI: 10.3150/22-BEJ1572

Abstract

We consider the i.i.d. Bernoulli field μp on Zd 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(1p)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.

Funding Statement

This work was funded by the German Research Foundation under Germany’s Excellence Strategy MATH+: The Berlin Mathematics Research Center, EXC-2046/1 project ID: 390685689 and the German Leibniz Association via the Leibniz Competition 2020.

Acknowledgements

The authors thank the anonymous referee for useful comments that helped to improve the manuscript. The authors also thank Nils Engler for inspiring discussions.

Citation

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Benedikt Jahnel. Christof Külske. "Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites." Bernoulli 29 (4) 3013 - 3032, November 2023. https://doi.org/10.3150/22-BEJ1572

Information

Received: 1 September 2021; Published: November 2023
First available in Project Euclid: 22 August 2023

MathSciNet: MR4632129
Digital Object Identifier: 10.3150/22-BEJ1572

Keywords: Bernoulli field , Dobrushin uniqueness , Gibbsianness , Gibbs-uniqueness , local thinning , Peierls’ argument , two-layer representation

Vol.29 • No. 4 • November 2023
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