Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Talagrand’s inequality for interacting particle systems satisfying a log-Sobolev inequality

Florian Völlering

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Abstract

Talagrand’s inequality for independent Bernoulli random variables is extended to many interacting particle systems (IPS). The main assumption is that the IPS satisfies a log-Sobolev inequality. In this context it is also shown that a slightly stronger version of Talagrand’s inequality is equivalent to a log-Sobolev inequality.

Additionally we also look at a common application, the relation between the probability of increasing events and the influences on that event by changing a single spin.

Résumé

Nous étendons l’inégalité de Talagrand pour des variables aléatoires de Bernoulli à une grande classe de systèmes de particules. L’hypothèse principale est que les systèmes de particules satisfont l’inégalité de log-Sobolev. Dans ce contexte nous démontrons également qu’une version plus forte de l’inégalité de Talagrand est équivalente à l’inégalité de log-Sobolev.

Nous considérons aussi comme application la relation entre la probablité d’un évènement croissant et les “influences” sur cet évènement du changement d’un seul spin.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 173-195.

Dates
Received: 4 December 2013
Revised: 15 May 2014
Accepted: 29 June 2014
First available in Project Euclid: 6 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089266

Digital Object Identifier
doi:10.1214/14-AIHP630

Mathematical Reviews number (MathSciNet)
MR3449300

Zentralblatt MATH identifier
1333.60210

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability

Keywords
Talagrand’s inequality Russo’s formula Variance estimate Dependent random variables Log-Sobolev inequality Interacting particle system

Citation

Völlering, Florian. Talagrand’s inequality for interacting particle systems satisfying a log-Sobolev inequality. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 173--195. doi:10.1214/14-AIHP630. https://projecteuclid.org/euclid.aihp/1452089266


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