logo AFST
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/
DO  - 10.5802/afst.1460
LA  - en
ID  - AFST_2015_6_24_4_691_0
ER  - 
%0 Journal Article
%A Pietro Caputo
%A Georg Menz
%A Prasad Tetali
%T Approximate tensorization of entropy at high temperature
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2015
%P 691-716
%V 24
%N 4
%I Université Paul Sabatier, Institut de Mathématiques
%C Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1460/
%R 10.5802/afst.1460
%G en
%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/

[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

[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 | Zbl

[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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

Cited by Sources: