Abstract
We consider Glauber dynamics reversible with respect to Gibbs measures with heavy tails in the case when spins are unbounded. The interactions are bounded and of finite range. The self-potential enters into two classes of measures, κ-concave probability measures and sub-exponential laws, for which it is known that no exponential decay can occur. Using coercive inequalities we prove that, for κ-concave probability measures, the associated infinite volume semi-group decays to equilibrium polynomially and stretched exponentially for sub-exponential laws. This improves and extends previous results by Bobkov and Zegarlinski.
References
Ané C., Blachère S., Chafaï D., Fougères P., Gentil I., Malrieu F., Roberto C., Scheffer G.: Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, vol. 10. Société Mathématique de France, Paris (2000)
Bakry, D.: L’hypercontractivité et son utilisation en théorie des semigroupes. In: Lectures on Probability theory. École d’été de Probabilités de St-Flour 1992. Lecture Notes in Mathematics, vol. 1581, pp. 1–114. Springer, Berlin (1994)
Barthe F., Cattiaux P., Roberto C.: Concentration for independent random variables with heavy tails. AMRX 2005(2), 39–60 (2005)
Barthe F., Cattiaux P., Roberto C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iber. 22(3), 993–1066 (2006)
Barthe F., Roberto C.: Sobolev inequalities for probability measures on the real line. Studia Math. 159(3), 481–497 (2003)
Barthe F., Roberto C.: Modified logarithmic Sobolev inequalities on \({\mathbb{R}}\) . Potential Analysis. 29(2), 167–193 (2008)
Barthe F., Kolesnikov A.V.: Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18(4), 921–979 (2008)
Bertini L., Cancrini N., Cesi F.: The spectral gap for a Glauber-type dynamics in a continuous gas. Ann. Inst. H. Poincaré Probab. Stat. 38(1), 91–108 (2002)
Bertini L., Zegarlinski B.: Coercive inequalities for Gibbs measures. J. Funct. Anal. 162(2), 257–286 (1999)
Bertini L., Zegarlinski B.: Coercive inequalities for Kawasaki dynamics. The product case. Markov Process. Relat. Fields 5(2), 125–162 (1999)
Bobkov S.G.: Large deviations and isoperimetry over convex probability measures. Electr. J. Prob. 12, 1072–1100 (2007)
Bobkov, S.G., Ledoux, M.: Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. (2007, in press)
Bobkov, S.G., Zegarlinski, B.: Distribution with slow tails and ergodicity of markov semigroups in infinite dimensions. Preprint (2008)
Borell C.: Convex set functions in d-space. Period. Math. Hungar. 6(2), 111–136 (1975)
Boudou A.-S., Caputo P., Dai Pra P., Posta G.: Spectral gap estimates for interacting particle systems via a Bochner-type identity. J. Funct. Anal. 232(1), 222–258 (2006)
Cancrini N., Caputo P., Martinelli F.: Relaxation time of L-reversal chains and other chromosome shuffles. Ann. Appl. Probab. 16(3), 1506–1527 (2006)
Cancrini N., Martinelli F., Roberto C., Toninelli C.: Kinetically constrained spin models. Probab. Theory Relat. Fields 140(3–4), 459–504 (2008)
Cattiaux, P., Gozlan, N., Guillin, A., Roberto, C.: Functional inequalities for heavy tails distributions and application to isoperimetry. Preprint (2008)
Cesi F.: Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probab. Theory Relat. Fields 120(4), 569–584 (2001)
Davies E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)
Dolbeault J., Gentil I., Guillin A., Wang F.Y.: l q functional inequalities and weighted porous media equations. Pot. Anal. 28(1), 35–59 (2008)
Georgii H.-O.: Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, vol. 9. Walter de Gruyter, Berlin (1988)
Gross, L.: Logarithmic Sobolev inequalities and contractivity properties of semi-groups. In: Dell’ Antonio, G., Mosco, U. (eds.) Dirichlet Forms. Lecture Notes in Mathematics, vol. 1563, pp. 54–88 (1993)
Guionnet, A., Zegarlinski, B.: Lectures on logarithmic Sobolev inequalities. Séminaire de Probabilités XXXVI. Lecture Notes in Mathematics, vol. 1801 (2002)
Landim C., Panizo G., Yau H.T.: Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems. Ann. Inst. H. Poincaré Probab. Stat. 38(5), 739–777 (2002)
Landim C., Yau H.T.: Convergence to equilibrium of conservative particle systems on \({\mathbb{Z}^{d}}\) . Ann. Probab. 31(1), 115–147 (2003)
Ledoux, M.: Concentration of measure and logarithmic Sobolev inequalities. In: Séminaire de Probabilités XXXIII. Lecture Notes in Mathematics, vol. 1709, pp. 120–216. Springer, Berlin (1999)
Liggett T.M.: L 2 rates of convergence for attractive reversible nearest particle systems: the critical case. Ann. Probab. 19(3), 935–959 (1991)
Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Mathematics, vol. 1717, pp 93–191. Springer, Berlin (1999)
Maz’ja V.G.: Sobolev spaces. Springer Series in Soviet Mathematics. Springer, Berlin (1985) (Translated from the Russian by T. O. Shaposhnikova)
Muckenhoupt B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972) (Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I)
Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Röckner M., Wang F.Y.: Weak Poincaré inequalities and L 2-convergence rates of Markov semigroups. J. Funct. Anal. 185(2), 564–603 (2001)
Royer G.: Une initiation aux inégalités de Sobolev logarithmiques. S.M.F., Paris (1999)
Talagrand, M.: A new isoperimetric inequality and the concentration of measure phenomenon. In: Geometric Aspects of Functional Analysis (1989–1990). Lecture Notes in Mathematics, vol. 1469, pp. 94–124. Springer, Berlin (1991)
Wang F.Y.: Functional Inequalities, Markov Processes and Spectral Theory. Science Press, Beijing (2005)
Wang F.Y.: Orlicz-Poincaré inequalities. Proc. Edinb. Math. Soc. 51(2), 529–543 (2008)
Wang F.-Y.: From super Poincaré to weighted log-Sobolev and entropy-cost inequalities. J. Math. Pures Appl. (9) 90(3), 270–285 (2008)
Zegarlinski B.: The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Commun. Math. Phys. 175(2), 401–432 (1996)
Zitt P.-A.: Functional inequalities and uniqueness of the Gibbs measure—from log-Sobolev to Poincaré. ESAIM Probab. Stat. 12, 258–272 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the European Research Council through the “Advanced Grant” PTRELSS 228032 and by GDRE 224 GREFI-MEFI, CNRS-INdAM.
Rights and permissions
About this article
Cite this article
Roberto, C. Slow decay of Gibbs measures with heavy tails. Probab. Theory Relat. Fields 148, 247–268 (2010). https://doi.org/10.1007/s00440-009-0229-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-009-0229-3
Keywords
- Weak Poincaré inequality
- Gibbs measures
- Glauber dynamics
- Unbounded spin systems
- Heavy tails distributions
Mathematics Subject Classification (2000)
- 60K35
- 82C22
- 26D10
- 47D07
- 82C20