Odd cutsets and the hard-core model on Z^d
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.
Cite
@article{arxiv.1106.3594,
title = {Odd cutsets and the hard-core model on Z^d},
author = {Ron Peled and Wojciech Samotij},
journal= {arXiv preprint arXiv:1106.3594},
year = {2017}
}
Comments
24 pages, 3 figures