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arXiv · 1106.3594

Odd cutsets and the hard-core model on Z^d

Ron Peled·Wojciech Samotij
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Abstract

We consider the hard-core lattice gas model on Z^d 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 Z^d, 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.

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Ron Peled, Wojciech Samotij. 2011-06-17. Odd cutsets and the hard-core model on Z^d. https://arxiv.org/abs/1106.3594

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