Odd cutsets and the hard-core model on \(\mathbb{Z}^{d}\)
type: Item
instance of: scholarly article
title: Odd cutsets and the hard-core model on \(\mathbb{Z}^{d}\)
zbMATH Open document ID: 1305.82019
DOI: 10.1214/12-AIHP535
author: Ron Peled
author: Wojciech Samotij
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.3594
full work available at URL: https://projecteuclid.org/euclid.aihp/1403277005
review text: The authors study the hardcore lattice gas on \(\mathbb Z^d\) and study in particular the high-dimensional, high-density situation, where there are multiple (Gibbs) measures. They improve on existing results by Galvin and Kahn, showing that there is a phase transition if the intensity parameter (``activity'') is larger than \(C d^{\frac{1}{3}} \log^2 d\), rather than the earlier known \(d^{\frac{1}{4}} (\log d)^{\frac{3}{4}}\). As the actual result is conjectured to behave like \(C d^{-1}\), possibly with logarithmic corrections, this result is not yet optimal, but the arguments are already quite ingenious. The authors use refined combinatorics on ``cutsets'' which play the roles of contours, separating the odd and even phases. Some of the arguments which they employ were earlier used by Peled.
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 82B41
zbMATH DE Number: 6340415
zbMATH Keywords: hard-core model
zbMATH Keywords: phase transition
zbMATH Keywords: high dimensions
zbMATH Keywords: cutsets
reviewed by: A. C. D. van Enter
MaRDI profile type: Publication
arXiv ID: 1106.3594
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description: scientific article; zbMATH DE number 6340415