Hypercontractivity for spin systems of infinite extension
type: Item
instance of: scholarly article
title: Hypercontractivity for spin systems of infinite extension
zbMATH Open document ID: 0820.60082
DOI: 10.1007/BF01192197
author: Étienne Laroche
publication date: 1995-04-23T00:00:00Z
review text: The author considers a spin system on \(Z^ d\), with values in a compact connected Riemannian manifold of finite dimension. He proves the equivalence between first a weak mixing condition, second the control of the spectral gap and third the control of logarithmic Sobolev constants, for not necessarily finite range Gibbs potentials. Consequently, he obtains informations about the \(L^ 2\)-decay of semigroups to the equilibrium and the decorrelation decay. There is no transitory rate between an algebraic decay as \(t^{-2d}\) (resp. \(| j - k|^{- 2d}\)), and an exponential decay. In a last part, the author considers discrete compact spins and proves the same equivalence result between the three above properties. In this particular situation, he deduces moreover that in case of phase transition (several Gibbs measures), the correlation between two spins at locations \(j\) and \(k\) cannot decrease uniformly more quickly than \(| j - k |^{-2d}\).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82C20
Mathematics Subject Classification ID: 82C26
zbMATH DE Number: 739105
zbMATH Keywords: spin system
zbMATH Keywords: infinite range Gibbs potential
zbMATH Keywords: Sobolev constant
zbMATH Keywords: spectral gap
zbMATH Keywords: mixing condition
reviewed by: Sylvie Méléard
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 739105