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The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice

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Abstract

Using a method based on the application of hypercontractivity we prove the strong exponential decay to equilibrium for a stochastic dynamics of unbounded spin system on a lattice.

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References

  • [AKR] Albeverio, S., Kondratjev, Y.G., Röckner, M.: Dirichlet Operators and Gibbs Measures. BiBoS Preprint 541, 1992

  • [BE] Bakry, D., Emery, M.: Diffusions hypercontractives. Sem. de Probabilities XIX, Azema, J., Yor, M. (eds.), Vol. LNm 1123, Berlin, Heidelberg, New York: Springer, 1984, pp. 177–206

    Google Scholar 

  • [BeH-K] Bellissard, J., Hoegh-Krohn, R.: Compactness and the maximal Gibbs state for random fields on a lattice. Commun. Math. Phys.84, 297–327 (1982)

    Google Scholar 

  • [Br] Brydges, D.C.: A Short Course on Cluster Expansions. Critical Phenomena, random systems, gauge theories, Osterwalder, K., Stora, R., Les Houches (eds.), vol.XLIII, 1984, pp. 129–183.

  • [CS] Carlen, E.A., Stroock, D.W.: An application of the Bakry-Emery criterion to infinite dimensional diffusion. Sem. de Probabilities XX, Azema, J., Yor, M. (eds.) LNm 1204, Berlin, Heidelberg, New York: Springer, pp. 341–348

  • [DaGSi] Davies, E.B., Gross, L., Simon, B.: Hypercontractivity: A bibliographical review. In: Proceedings of the Hoegh-Krohn Memorial Conference

  • [DaSi] Davies, E.B., Simon, B.: Ultracontractivity and the Heat Kernel for Schrödinger Operators and Dirichlet Laplacians. J. Func. Anal.59, 335–395 (1984)

    Google Scholar 

  • [Di] Dimock, J.: A Cluster Expansion for Stochastic Lattice Fields. J. Stat. Phys.58, 1181–1207 (1990)

    Google Scholar 

  • [GJS] Glimm, J., Jaffe, A., Spencer, T.: The Particle Structure of the weakly-coupled P(ϕ)2 model and other applications of high-temperature expansions. Constructive Field Theory, Velo, G., Wightman, A.S. (eds.), Berlin, Heidelberg, New York: Springer, 1973

    Google Scholar 

  • [G] Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math.97, 1061–1083 (1976)

    Google Scholar 

  • [HS] Holley, R., Stroock, D.W.: Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys.46, 1159–1194 (1987)

    Google Scholar 

  • [LY] Lu, ShengLin, Yau, Horng-Tzer: Spectral Gap and Logarithmic Sobolev Inequality for Kawasaki and Glauber Dynamics. Commun. Math. Phys.156, 399–433 (1993)

    Google Scholar 

  • [MO] Martinelli, F., Olivieri, E.: Approach to Equilibrium of Glauber Dynamics in the One Phase Region: I. The Attractive case II. The General Case. Commun. Math. Phys.161, 447–486/487–514 (1994)

    Google Scholar 

  • [MM] Malyshev, W.A., Minlos, R.A.: Gibbsian Random Fields: The Method of Cluster Expansion. Moscow: Nauka, 1985

    Google Scholar 

  • [R] Rosen, J.: Sobolev Inequalities for Weight Spaces and Supercontractivity. Trans. A.M.S.222, 367–376 (1976)

    Google Scholar 

  • [Rot] Rothaus, O.S.: Logarithmic Sobelev Inequalities and the Spectrum of Schrödinger Operators. J. Funct. Anal.42, 110–378 (1981)

    Google Scholar 

  • [Si1] Simon, B.: A remark on Nelson's best hypercontractive estimates. Proc. A.M.S.55, 376–378 (1976)

    Google Scholar 

  • [Si2] Simon, B.: The:P(ϕ)2 Euclidean (Quantum) Field Theory. Princeton, NJ: Princeton University Press, 1974

    Google Scholar 

  • [SZ1] Stroock, D.W., Zegarlinski, B.: The Logarithmic Sobolev Inequality for Continuous Spin Systems on a Lattice. J. Funct. Anal.104, 299–326 (1992)

    Google Scholar 

  • [SZ2] Stroock, D.W., Zegarlinski, B.: The Equivalence of the Logarithmic Sobolev Inequality and the Dobrushin-Shlosman Mixing Condition. Commun. Math. Phys.144, 303–323 (1992)

    Google Scholar 

  • [SZ3] Stroock, D.W., Zegarlinski, B.: The Logarithmic Sobolev Inequality for Discrete Spin Systems on a Lattice. Commun. Math. Phys.149, 175–193 (1992)

    Google Scholar 

  • [SZ4] Stroock, D.W., Zegarlinski, B.: On the Ergodic Properties of Glauber Dynamics. Preprint

  • [Z1] Zegarlinski, B.: On log-Sobolev inequalities for infinite lattice systems. Lett. Math. Phys.20, 173–182 (1990)

    Google Scholar 

  • [Z2] Zegarlinski, B.: Log-Sobolev inequalities for infinite one-dimensional lattice systems. Commun. Math. Phys.133, 147–162 (1990)

    Google Scholar 

  • [Z3] Zegarlinski, B.: Dobrushin uniqueness theorem and logarithmic Sobolev inequalities. J. Funct. Anal.105, 77–111 (1992)

    Google Scholar 

  • [Z4] Zegarlinski, B.: Hypercontractive Markov Semigroups. Unpublished lecture notes 1992

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Authors and Affiliations

  1. Mathematics Department, Imperial College, SW7 2BZ, London, UK

    Boguslaw Zegarlinski

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  1. Boguslaw Zegarlinski
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Communicated by D.C. Brydges

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Zegarlinski, B. The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Commun.Math. Phys. 175, 401–432 (1996). https://doi.org/10.1007/BF02102414

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  • DOI: https://doi.org/10.1007/BF02102414

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