Odd cutsets and the hard-core model on ℤd

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Abstract

We consider the hard-core lattice gas model on ℤd and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceeds Cd-1/3(log d)2, the model exhibits multiple hard-core measures, thus improving the previous bound of Cd-1/4(log d)3/4 given by Galvin and Kahn. At the heart of our approach lies the study of a certain class of edge cutsets in ℤd, the so-called odd cutsets, that appear naturally as the boundary between different phases in the hard-core model. We provide a refined combinatorial analysis of the structure of these cutsets yielding a quantitative form of concentration for their possible shapes as the dimension d tends to infinity. This analysis relies upon and improves previous results obtained by the first author.

Original languageEnglish
Pages (from-to)975-998
Number of pages24
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume50
Issue number3
DOIs
StatePublished - Aug 2014

Keywords

  • Edge cutsets
  • Gibbs measures
  • Hard-core model
  • Integer lattice

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