Odd cutsets and the hard-core model on $\mathbb {Z}^{d}$

Peled, Ron; Samotij, Wojciech

doi:10.1214/12-AIHP535

Odd cutsets and the hard-core model on

ℤ^{d}

Peled, Ron ; Samotij, Wojciech

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 975-998.

Résumé
Abstract

Nous étudions la structure de phase, en grande dimension, d’un modèle de sphères dures sur le réseau $ℤ^{d}$ . Nous prouvons que le modèle présente plusieurs mesures lorsque le paramètre de densité dépasse $C d^{- 1 / 3} {(log d)}^{2}$ , améliorant ainsi la borne de $C d^{- 1 / 4} {(log d)}^{3 / 4}$ obtenue par Galvin et Kahn. Notre approche repose sur l’étude de certaines classes d’ensembles séparateurs dans $ℤ^{d}$ , constituées d’ensembles impaires, qui délimitent la frontière entre différentes phases du modèle de sphères dures. Nous faisons une analyse combinatoire précise de la structure de ces ensembles séparateurs et obtenons une forme quantitative de la concentration des différentes formes possibles prises par ces ensembles lorsque la dimension $d$ tend vers l’infini. Cette analyse repose sur des méthodes obtenues auparavant par le premier auteur, tout en les améliorant.

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.

MR Zbl

DOI : 10.1214/12-AIHP535

Classification : 82B20, 60C05, 05C30, 05A16
Mots clés : edge cutsets, Gibbs measures, hard-core model, integer lattice

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     author = {Peled, Ron and Samotij, Wojciech},
     title = {Odd cutsets and the hard-core model on $\mathbb {Z}^{d}$},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {975--998},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     doi = {10.1214/12-AIHP535},
     mrnumber = {3224296},
     zbl = {1305.82019},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/12-AIHP535/}
}

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Peled, Ron; Samotij, Wojciech. Odd cutsets and the hard-core model on $\mathbb {Z}^{d}$. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 975-998. doi : 10.1214/12-AIHP535. http://www.numdam.org/articles/10.1214/12-AIHP535/

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