The Annals of Probability
- Ann. Probab.
- Volume 34, Number 5 (2006), 1960-1989.
Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition
Full-text: Open access
Abstract
In in this paper we establish an explicit and sharp estimate of the spectral gap (Poincaré inequality) and the transportation inequality for Gibbs measures, under the Dobrushin uniqueness condition. Moreover, we give a generalization of the Liggett’s M−ɛ theorem for interacting particle systems.
Article information
Source
Ann. Probab. Volume 34, Number 5 (2006), 1960-1989.
Dates
First available in Project Euclid: 14 November 2006
Permanent link to this document
https://projecteuclid.org/euclid.aop/1163517230
Digital Object Identifier
doi:10.1214/009117906000000368
Mathematical Reviews number (MathSciNet)
MR2271488
Zentralblatt MATH identifier
1111.60079
Subjects
Primary: 60E15: Inequalities; stochastic orderings 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G60: Random fields
Keywords
Poincaré inequality transportation inequality Gibbs measure Dobrushin’s uniqueness condition
Citation
Wu, Liming. Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition. Ann. Probab. 34 (2006), no. 5, 1960--1989. doi:10.1214/009117906000000368. https://projecteuclid.org/euclid.aop/1163517230.
References
- Bakry, D. and Emery, M. (1985). Diffusions hypercontractives. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123 177--206. Springer, Berlin. Mathematical Reviews (MathSciNet): MR0889476
Zentralblatt MATH: 0561.60080
Digital Object Identifier: doi:10.1007/BFb0075847 - Bobkov, S. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1--28. Mathematical Reviews (MathSciNet): MR1682772
Digital Object Identifier: doi:10.1006/jfan.1998.3326
Zentralblatt MATH: 0924.46027 - Cassandro, M., Olivieri, E., Pellegrinotti, A. and Presutti, E. (1977/78). Existence and uniqueness of DLR measures for unbounded spin systems. Z. Wahrsch. Verw. Gebiete 41 313--334. Mathematical Reviews (MathSciNet): MR0471115
Digital Object Identifier: doi:10.1007/BF00533602
Zentralblatt MATH: 0369.60116 - Chen, M. F. (2005). Eigenvalues, Inequalities, and Ergodic Theory. Springer, London.
- Chen, M. F. (2004). Spectral gap of Gibbs measures and phase transition. Preprint.
- Chen, M. F. and Wang, F. Y. (1997). General formula for lower bound of the first eigenvalue on Riemannian manifolds. Sci. China Ser. A 40 384--394.
- Djellout, H., Guillin, A. and Wu, L. M. (2004). Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 2702--2732. Mathematical Reviews (MathSciNet): MR2078555
Digital Object Identifier: doi:10.1214/009117904000000531
Project Euclid: euclid.aop/1091813628
Zentralblatt MATH: 1061.60011 - Dobrushin, R. L. (1968). The description of a random field by means of conditional probabilities and condition of its regularity. Theory Probab. Appl. 13 197--224. Mathematical Reviews (MathSciNet): MR0231434
- Dobrushin, R. L. (1970). Prescribing a system of random variables by conditional distributions. Theory Probab. Appl. 15 458--486.
- Dobrushin, R. L. and Shlosman, S. B. (1987). Completely analytical interactions: Constructive description. J. Statist. Phys. 46 983--1014. Mathematical Reviews (MathSciNet): MR0893129
Digital Object Identifier: doi:10.1007/BF01011153
Zentralblatt MATH: 0683.60080 - Dyson, F. L., Lieb, E. H. and Simon, B. (1978). Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Statist. Phys. 18 335--383.
- Föllmer, H. (1982). A covariance estimate for Gibbs measure. J. Funct. Anal. 46 387--395. Mathematical Reviews (MathSciNet): MR0661878
Digital Object Identifier: doi:10.1016/0022-1236(82)90053-2
Zentralblatt MATH: 0482.60052 - Georgii, H. O. (1988). Gibbs Measures and Phase Transition. de Gruyter, Berlin.
- Glimm, J. and Jaffe, A. (1987). Quantum Physics---A Functional Integral Point of View, 2nd ed. Springer, New York. Mathematical Reviews (MathSciNet): MR0887102
- Gozlan, N. (2004). Principe conditionnel de Gibbs pour des contraintes fines approchées et inégalités de transport. Thèse de Doctorat, Univ. Paris X.
- Guionnet, A. and Zegarlinski, B. (2003). Lectures on logarithmic Sobolev inequalities. Séminaire de Probabilités XXXVI. Lecture Notes in Math. 1801 1--134. Springer, Berlin.
- Helffer, B. (1999). Remarks on decay of correlations and Witten Laplacians III: Application to logarithmic Sobolev inequalities. Ann. Inst. H. Poincaré Probab. Statist. 35 483--508. Mathematical Reviews (MathSciNet): MR1702239
Digital Object Identifier: doi:10.1016/S0246-0203(99)00103-X - Ledoux, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709 120--216. Springer, Berlin.
- Ledoux, M. (2001). Logarithmic Sobolev inequalities for unbounded spin systems revisited. Séminaire de Probabilités. Lecture Notes in Math. 1755 167--194. Springer, Berlin.
- Levin, S. L. (1980). Application of Dobrushin's uniqueness theorem to $N$-vector models. Comm. Math. Phys. 78 65--74. Mathematical Reviews (MathSciNet): MR0597030
Digital Object Identifier: doi:10.1007/BF01941969
Project Euclid: euclid.cmp/1103908501 - Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.
- Maes, Ch. and Shlosman, S. B. (1993). When is an interacting particle system ergodic? Comm. Math. Phys. 151 447--466. Mathematical Reviews (MathSciNet): MR1207259
Digital Object Identifier: doi:10.1007/BF02097021
Project Euclid: euclid.cmp/1104252233
Zentralblatt MATH: 0765.60102 - Martinelli, F. (1999). Lectures on Glauber dynamics for discrete spin models. Ecole d'Eté de Saint-Flour. Lecture Notes in Math. 1717 93--191. Springer, Berlin.
- Marton, K. (1997). A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 556--571.
- Marton, K. (2004). Measure concentration for Euclidean distance in the case of dependent random variables. Ann. Probab. 32 2526--2544. Mathematical Reviews (MathSciNet): MR2078549
Digital Object Identifier: doi:10.1214/009117904000000702
Project Euclid: euclid.aop/1091813622 - Massart, P. (2003). Concentration inequalities and model selection. Available at http://www.math.u-psud.fr/~massart/stf2003_massart.pdf.
- Minlos, R. A. and Trishch, A. G. (1994). Complete spectral resolution of the generator of Glauber dynamics for the one-dimensional Ising model. Uspekhi Mat. Nauk 49 209--210. (Translation in Russian Math. Surveys 49 210--211.) Mathematical Reviews (MathSciNet): MR1316879
- Salas, J. and Sokal, A. D. (1996). Absense of phase transition for anti-ferromagnetic Potts models via the Dobrushin uniqueness theorem. J. Statist. Phys. 86 551--579. Mathematical Reviews (MathSciNet): MR1438965
Digital Object Identifier: doi:10.1007/BF02199113
Zentralblatt MATH: 0935.82010 - Simon, B. (1979). A remark on Dobrushin's uniqueness theorem. Comm. Math. Phys. 68 183--185. Mathematical Reviews (MathSciNet): MR0543198
Digital Object Identifier: doi:10.1007/BF01418127
Project Euclid: euclid.cmp/1103905311
Zentralblatt MATH: 0435.60099 - Stroock, D. W. and Zegarlinski, B. (1992). The equivalence between the logarithmic Sobolev inequality and the Dobrushin--Shlosman mixing condition. Comm. Math. Phys. 144 303--323. Mathematical Reviews (MathSciNet): MR1152374
Digital Object Identifier: doi:10.1007/BF02101094
Project Euclid: euclid.cmp/1104249318
Zentralblatt MATH: 0745.60104 - Stroock, D. W. and Zegarlinski, B. (1992). The logarithmic Sobolev inequality for discrete spin systems on the lattice. Comm. Math. Phys. 149 175--193. Mathematical Reviews (MathSciNet): MR1182416
Digital Object Identifier: doi:10.1007/BF02096629
Project Euclid: euclid.cmp/1104251144
Zentralblatt MATH: 0758.60070 - Villani, C. (2003). Topics in Optimal Transportation. Amer. Math. Soc., Providence, RI.
- Wick, W. D. (1981). Convergence to equilibrium of the stochastic Heisenberg model. Comm. Math. Phys. 81 361--377. Mathematical Reviews (MathSciNet): MR0634159
Digital Object Identifier: doi:10.1007/BF01209073
Project Euclid: euclid.cmp/1103920323
Zentralblatt MATH: 0474.60086 - Wu, L. M. (1995). Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23 420--445. Mathematical Reviews (MathSciNet): MR1330777
Digital Object Identifier: doi:10.1214/aop/1176988393
Project Euclid: euclid.aop/1176988393
Zentralblatt MATH: 0828.60017 - Wu, L. M. (2004). Essential spectral radius for Markov semigroups. I. Discrete time case. Probab. Theory Related Fields 128 255--321. Mathematical Reviews (MathSciNet): MR2031227
Digital Object Identifier: doi:10.1007/s00440-003-0304-0
Zentralblatt MATH: 1056.60068 - Wu, L. M. (2004). Estimate of spectral gap for continuous gas. Ann. Inst. H. Poincaré Probab. Statist. 40 387--409. Mathematical Reviews (MathSciNet): MR2070332
Digital Object Identifier: doi:10.1016/j.anihpb.2003.11.003 - Yoshida, N. (2001). The equivalence of the logarithmic Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Ann. Inst. H. Poincaré Probab. Statist. 37 223--243. Mathematical Reviews (MathSciNet): MR1819124
Digital Object Identifier: doi:10.1016/S0246-0203(00)01066-9 - Zegarliski, B. (1992). Dobrushin uniqueness theorem and logarithmic Sobolev inequalities. J. Funct. Anal. 105 77--111. Mathematical Reviews (MathSciNet): MR1156671
Digital Object Identifier: doi:10.1016/0022-1236(92)90073-R - Zhang, S. Y. (1999). Existence of optimal measurable coupling and ergodicity for Markov processes. Sci. China Ser. A 42 58--67.
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