Odd cutsets and the hard-core model on ℤ d

Ron Peled; Wojciech Samotij

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 3, page 975-998
  • ISSN: 0246-0203

Abstract

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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 C d - 1 / 3 ( log d ) 2 , the model exhibits multiple hard-core measures, thus improving the previous bound of C d - 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.

How to cite

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Peled, Ron, and Samotij, Wojciech. "Odd cutsets and the hard-core model on $\mathbb {Z}^{d}$." Annales de l'I.H.P. Probabilités et statistiques 50.3 (2014): 975-998. <http://eudml.org/doc/272020>.

@article{Peled2014,
abstract = {We consider the hard-core lattice gas model on $\mathbb \{Z\}^\{d\}$ 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 $\mathbb \{Z\}^\{d\}$, 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.},
author = {Peled, Ron, Samotij, Wojciech},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {edge cutsets; Gibbs measures; hard-core model; integer lattice; phase transition; high dimensions; cutsets},
language = {eng},
number = {3},
pages = {975-998},
publisher = {Gauthier-Villars},
title = {Odd cutsets and the hard-core model on $\mathbb \{Z\}^\{d\}$},
url = {http://eudml.org/doc/272020},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Peled, Ron
AU - Samotij, Wojciech
TI - Odd cutsets and the hard-core model on $\mathbb {Z}^{d}$
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 3
SP - 975
EP - 998
AB - We consider the hard-core lattice gas model on $\mathbb {Z}^{d}$ 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 $\mathbb {Z}^{d}$, 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.
LA - eng
KW - edge cutsets; Gibbs measures; hard-core model; integer lattice; phase transition; high dimensions; cutsets
UR - http://eudml.org/doc/272020
ER -

References

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