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Approximate tensorization of entropy at high temperature
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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.

Published online : 2016-01-21
@article{AFST_2015_6_24_4_691_0,
     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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {4},
     year = {2015},
     pages = {691-716},
     language = {en},
     url={afst.centre-mersenne.org/item/AFST_2015_6_24_4_691_0/}
}
Caputo, Pietro; Menz, Georg; Tetali, Prasad. 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. https://afst.centre-mersenne.org/item/AFST_2015_6_24_4_691_0/

[1] Ané (C.), Blachère (S.), Chafaï(D.), Fougères (P.), Gentil (I.), Malrieu (F.), Roberto (C.), and Scheffer (G.).— Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris (2000). With a preface by Dominique Bakry and Michel Ledoux. | Zbl 0982.46026

[2] Bakry (D.) and Émery (M.).— Diffusions hypercontractives. In Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math., p. 177-206. Springer, Berlin (1985). | Numdam | MR 889476 | Zbl 0561.60080

[3] Bakry (D.), /sc Gentil (I.), and Ledoux (M.).— Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der Mathematischen Wissenschaften. Springer, Cham (2014). | MR 3155209

[4] Balister (P.) and Bollobás (B.).— Projections, entropy and sumsets. Combinatorica, 32(2) p. 125-141 (2012). | MR 2927635 | Zbl 1299.60013

[5] Bobkov (S. G.) and Tetali (P.).— Modified logarithmic Sobolev inequalities in discrete settings.J. Theoret. Probab., 19(2) p. 289-336 (2006). | MR 2283379 | Zbl 1113.60072

[6] Bodineau (T.) and Helffer (B.).— The log-Sobolev inequality for unbounded spin systems. J. Funct. Anal., 166(1) p. 168-178 (1999). | MR 1704666 | Zbl 0972.82035

[7] Caputo (P.), Dai Pra (P.), and Posta (G.).— Convex entropy decay via the Bochner-Bakry-Emery approach. Ann. Inst. Henri Poincaré Probab. Stat., 45(3) p. 734-753 (2009). | Numdam | MR 2548501 | Zbl 1181.60142

[8] Caputo (P.) and Posta (G.).— Entropy dissipation estimates in a zero-range dynamics. Probab. Theory Related Fields, 139(1-2) p. 65-87 (2007). | MR 2322692 | Zbl 1126.60082

[9] Carlen (E. A.), Lieb (E. H.), and Loss (M.).— A sharp analog of Young’s inequality on S N and related entropy inequalities. J. Geom. Anal., 14(3) p. 487-520 (2004). | MR 2077162 | Zbl 1056.43002

[10] Carlen (E. A.), Lieb (E. H.), and Loss (M.).— An inequality of Hadamard type for permanents. Methods Appl. Anal., 13(1) p. 1-17 (2006). | MR 2275869 | Zbl 1116.15015

[11] Cesi (F.).— Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probab. Theory Related Fields, 120(4) p. 569-584 (2001). | MR 1853483 | Zbl 1086.82002

[12] Chafaï (D.) and Joulin (A.).— Intertwining and commutation relations for birth-death processes. Bernoulli, 19(5A) p. 1855-1879 (2013). | MR 3129037 | Zbl 1286.60084

[13] Dai Pra (P.), Paganoni (A. M.), and Posta (G.).— Entropy inequalities for unbounded spin systems. Ann. Probab., 30(4) p. 1959-1976 (2002). | MR 1944012 | Zbl 1013.60076

[14] Dai Pra (P.) and Posta (G.).— Entropy decay for interacting systems via the Bochner-Bakry-Émery approach. Electron. J. Probab., 18, no. 52, 21 (2013). | MR 3065862 | Zbl 1286.60097

[15] Diaconis (P.) and Saloff-Coste (L.).— Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab., 6(3) p. 695-750 (1996). | MR 1410112 | Zbl 0867.60043

[16] Ding (J.), Lubetzky (E.), and Peres (Y.).— The mixing time evolution of Glauber dynamics for the meanfield Ising model. Comm. Math. Phys., 289(2) p. 725-764 (2009). | MR 2506768 | Zbl 1173.82018

[17] Erbar (M.) and Maas (J.).— Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal., 206(3) p. 997-1038 (2012). | MR 2989449 | Zbl 1256.53028

[18] Gao (F.) and Quastel (J.).— Exponential decay of entropy in the random transposition and Bernoulli-Laplace models. Ann. Appl. Probab., 13(4) p. 1591-1600 (2003). | MR 2023890 | Zbl 1046.60003

[19] Goel (S.).— Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl., 114(1) p. 51-79 (2004). | MR 2094147 | Zbl 1074.60080

[20] Guionnet (A.) and Zegarlinski (B.).— Lectures on logarithmic Sobolev inequalities. In Séminaire de Probabilités, XXXVI, volume 1801 of Lecture Notes in Math., pages 1-134. Springer, Berlin (2003). | Numdam | MR 1971582 | Zbl 1125.60111

[21] Holley (R.) and Stroock (D.).— Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys., 46(5-6) p. 1159-1194 (1987). | MR 893137 | Zbl 0682.60109

[22] Johnson (O.).— Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl., 117(6) p. 791-802 (2007). | MR 2327839 | Zbl 1115.60012

[23] Lu (S. L.) and Yau (H.-T.).— Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys., 156(2) p. 399-433 (1993). | MR 1233852 | Zbl 0779.60078

[24] Madiman (M.) and Tetali (P.).— Information inequalities for joint distributions, with interpretations and applications. IEEE Trans. Inform. Theory, 56(6) p. 2699-2713 (2010). | MR 2683430

[25] Martinelli (F.) and Olivieri (E.).— Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Comm. Math. Phys., 161(3) p. 487-514 (1994). | MR 1269388 | Zbl 0793.60111

[26] Martinelli (F.).— Lectures on Glauber dynamics for discrete spin models. In Lectures on probability theory and statistics (Saint-Flour, 1997), volume 1717 of Lecture Notes in Math., p. 93-191. Springer, Berlin (1999). | MR 1746301 | Zbl 0930.00052

[27] Marton (K.).— An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces. J. Funct. Anal., 264(1) p. 34-61 (2013). | MR 2995699 | Zbl 1260.60030

[28] Montenegro (R.) and Tetali (P.).— Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci., 1(3) p. x+121 (2006). | MR 2341319 | Zbl 1193.68138

[29] Otto (F.) and Reznikoff (M. G.).— A new criterion for the logarithmic Sobolev inequality and two applications. J. Funct. Anal., 243(1) p. 121-157 (2007). | MR 2291434 | Zbl 1109.60013

[30] Stroock (D. W.) and Zegarlinski (B.).— The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys., 149(1) p. 175-193 (1992). | MR 1182416 | Zbl 0758.60070

[31] Yoshida (N.).— The log-Sobolev inequality for weakly coupled lattice fields. Probab. Theory Related Fields, 115(1) p. 1-40 (1999). | MR 1715549 | Zbl 0948.60095

[32] Zegarlinski (B.).— On log-Sobolev inequalities for infinite lattice systems. Lett. Math. Phys., 20(3) p. 173-182 (1990). | MR 1074698 | Zbl 0717.47015

[33] Zegarlinski (B.).— Dobrushin uniqueness theorem and logarithmic Sobolev inequalities. J. Funct. Anal., 105(1) p. 77-111 (1992). | MR 1156671 | Zbl 0761.46020

[34] Zegarlinski (B.).— The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Comm. Math. Phys., 175(2) p. 401-432 (1996). | MR 1370101 | Zbl 0844.46050