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- Série 6 : Tome 24 (2015)

- no. 4

- p. 691-716

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We show that for weakly dependent random variables the relative entropy functional satisfies an approximate version of the standard tensorization property which holds in the independent case. As a corollary we obtain a family of dimensionless logarithmic Sobolev inequalities. In the context of spin systems on a graph, the weak dependence requirements resemble the well known Dobrushin uniqueness conditions. Our results can be considered as a discrete counterpart of a recent work of Katalin Marton [27]. We also discuss some natural generalizations such as approximate Shearer estimates and subadditivity of entropy.

On montre que pour des variables aléatoires faiblement dépendentes l’entropie relative satisfait une version approximée de la propriété de tensorisation associée au cas indépendent. Cela implique une famille d’inégalités de Sobolev logarithmiques indépendentes de la dimension. Pour des systèmes de spin en interaction sur un graphe, la condition de dépendence faible devient une sorte de condition de unicité de Dobrushin. Nos résultats représentent par ailleurs une version discrète d’un travail récent par Katalin Marton [27]. On considère aussi des généralisations naturelles de ces résultats tels que des inégalités de Shearer approximées.

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Pietro Caputo 1 ; Georg Menz 2 ; Prasad Tetali 3

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