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Approximate tensorization of entropy at high temperature
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 4, pp. 691-716.

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.

DOI: 10.5802/afst.1460
Pietro Caputo 1; Georg Menz 2; Prasad Tetali 3

1 Università Roma Tre.
2 Stanford University
3 Georgia Institute of Technology
     author = {Pietro Caputo and Georg Menz and Prasad Tetali},
     title = {Approximate tensorization of entropy at high temperature},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {691--716},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 24},
     number = {4},
     year = {2015},
     doi = {10.5802/afst.1460},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1460/}
AU  - Pietro Caputo
AU  - Georg Menz
AU  - Prasad Tetali
TI  - Approximate tensorization of entropy at high temperature
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2015
SP  - 691
EP  - 716
VL  - 24
IS  - 4
PB  - Université Paul Sabatier, Institut de Mathématiques
PP  - Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1460/
UR  - https://doi.org/10.5802/afst.1460
DO  - 10.5802/afst.1460
LA  - en
ID  - AFST_2015_6_24_4_691_0
ER  - 
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%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2015
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%I Université Paul Sabatier, Institut de Mathématiques
%C Toulouse
%U https://doi.org/10.5802/afst.1460
%R 10.5802/afst.1460
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%F AFST_2015_6_24_4_691_0
Pietro Caputo; Georg Menz; Prasad Tetali. Approximate tensorization of entropy at high temperature. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 4, pp. 691-716. doi : 10.5802/afst.1460. https://afst.centre-mersenne.org/articles/10.5802/afst.1460/

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