Lower bounds on the cluster size distribution

Michael Aizenman; Fran Çois Delyon; Bernard Souillard

doi:10.1007/BF01011369

Lower bounds on the cluster size distribution

Michael Aizenman
, Fran Çois Delyon
, Bernard Souillard

Research output: Contribution to journal › Article › peer-review

27 Scopus citations

Abstract

We rigorously prove that the probability P_n that the origin of a d-dimensional lattice belongs to a cluster of exactly n sites satisfies P_n > exp(-αn^(d-1)/d) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentration p > p_c, as well as for the equilibrium states of lattice spin systems with quite general interactions. Such a lower bound applies also if no percolation occurs, but if it appears in some other phase of the system.

Original language	English (US)
Pages (from-to)	267-280
Number of pages	14
Journal	Journal of Statistical Physics
Volume	23
Issue number	3
DOIs	https://doi.org/10.1007/BF01011369
State	Published - Sep 1980
Externally published	Yes

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

Keywords

Gibbs states
Percolation
cluster size distribution
nucleation
stochastic geometry

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10.1007/BF01011369

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abstract = "We rigorously prove that the probability Pn that the origin of a d-dimensional lattice belongs to a cluster of exactly n sites satisfies Pn > exp(-αn(d-1)/d) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentration p > pc, as well as for the equilibrium states of lattice spin systems with quite general interactions. Such a lower bound applies also if no percolation occurs, but if it appears in some other phase of the system.",

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AU - Aizenman, Michael

AU - Delyon, Fran Çois

AU - Souillard, Bernard

PY - 1980/9

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N2 - We rigorously prove that the probability Pn that the origin of a d-dimensional lattice belongs to a cluster of exactly n sites satisfies Pn > exp(-αn(d-1)/d) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentration p > pc, as well as for the equilibrium states of lattice spin systems with quite general interactions. Such a lower bound applies also if no percolation occurs, but if it appears in some other phase of the system.

AB - We rigorously prove that the probability Pn that the origin of a d-dimensional lattice belongs to a cluster of exactly n sites satisfies Pn > exp(-αn(d-1)/d) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentration p > pc, as well as for the equilibrium states of lattice spin systems with quite general interactions. Such a lower bound applies also if no percolation occurs, but if it appears in some other phase of the system.

KW - Gibbs states

KW - Percolation

KW - cluster size distribution

KW - nucleation

KW - stochastic geometry

UR - https://www.scopus.com/pages/publications/0039832420

UR - https://www.scopus.com/pages/publications/0039832420#tab=citedBy

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SP - 267

EP - 280

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3

ER -

Lower bounds on the cluster size distribution

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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