Abstract
Various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed. In particular somefinite size conditions are discussed, together with their implications for the Gibbs measures and for the approach to equilibrium of Glauber dynamics inarbitrarily large volumes. It is shown that Dobrushin-Shlosman's theory ofcomplete analyticity and its dynamical counterpart due to Stroock and Zegarlinski, cannot be applied, in general, to the whole one phase region since it requires mixing properties for regions ofarbitrary shape. An alternative approach, based on previous ideas of Oliveri, and Picco, is developed, which allows to establish results on rapid approach to equilibrium deeply inside the one phase region. In particular, in the ferromagnetic case, we considerably improve some previous results by Holley and Aizenman and Holley. Our results are optimal in the sene that, for example, they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed-dimensional Ising model with or without an external field. In part II we extensively consider the general case (not necessarily attractive) and we develop a new method, based on renormalizations group ideas and on an assumption of strong mixing in a finite cube, to prove hypercontractivity of the Markov semigroup of the Glauber dynamics.
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[AH] Aizenman, M., Holley, R.: Rapid convergence to equilibrium of stochastic ising models in the Dobrushin-Shlosman regime percolation theory and ergodic theory of infinite particle systems. H. Kesten (ed.) IMS Volumes in Math. and Appl., Berlin Heidelberg, New York: Springer 1987, pp. 1–11.
[D1] Dobrushin, R.L.: Private communciation
[D2] Dobrushin, R.L.: Prescribing a System of random variables by conditional distributions. Theory of Prob. Appl.15, 453–486 (1970)
[DP] Dobrushin, R.L., Pecherski, E.A.: Uniqueness condition for finitely dependent random fields. In: Random fields. Amsterdam-Oxford-New York: North-Holland,1, 223–262 (1981)
[DS1] Dobrushin, R.L., Shlosman, S.: Constructive criterion for the uniqueness of Gibbs fields. Stat. Phys. and Dyn. Syst., Birkhäuser 1985, pp. 347–370
[DS2] Dobrushin, R.L, Shlosman, S.: Completely analytical. Gibbs fields. Stat. Phys. and Dyn. Syst., Birkhäuser 1985, pp. 371–403
[DS3] Dobrushin, R.L., Shlosman, S.: Completely analytical interactions constructive description. J. Stat. Phys.46, 983–1014 (1987)
[EFS] v. Enter, A., Fernandez, R., Sokal, A.: Regularity properties and pathologies of position-space renormalization-group transformations. To appear in J. Stat. Phys. 1992
[FKG] Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89–103 (1971)
[H1] Holley, R.: Remarks on the FKG inequalities. Commun. Math. Phys.36, 227–231 (1976)
[H2] Holley, R.: Possible rates of convergence in finite range attraction spin systems. Contemp. Math.41, 215–234 (1985)
[Hi] Higuchi: Coexistence of infinite*-cluster. Ising percolation in two dimensions. Preprint. Kobe Univ., 1992
[HS] Holley, R., Stroock, D.W.: Application of the stochastic Ising model to Gibbs states. Commun. Math. Phys.48, 246–265 (1976)
[L] Ligget, T.: Interacting particle systems. Berlin, Heidelberg, New York: Springer 1985
[M] Martirosyan, D.G.: Theorems on strips in the classical Ising ferromagnetic model. Sov. J. Contemp. Math.22, 59–83 (1987)
[MO] Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Commun. Math. Phys.161, 487–514 (1994)
[MOS] Martinelli, F., Olivieri, E., Scoppola, E.: Metastability and exponential approach to equilibrium for low temperature stochastic Ising models. J. Stat. Phys.61, 1105–1119 (1990)
[MOSh] Martinelli, F., Olivieri, E., Schonmann, R.: For Gibbs state of 2D lattice spin systems weak mixing implies strong mixing. Commun. Math. Phys. in press
[NS] Neves, E.J., Schonmann, R.H.: Critical droplets and metastability for a Glauber dynamics at very low temperature. Commun. Math. Phys.137, 209–230 (1991)
[O] Olivieri, E.: On a cluster expansion for lattice spin systems a finite size condition for the convergence. J. Stat. Phys.50, 1179–1200 (1988)
[OP] Olivieri, E., Picco, P.: Clustering for D-dimensional lattice systems and finite volume factorization propertie. J. Stat. Phys.59, 221–256 (1990)
[S] Schonmann, R.: Private communication
[Sh] Shlosman, S.B.: Uniqueness and half-space nonuniqueness of Gibbs states in Czech models. Theor. Math. Phys.66, 284–293 (1986)
[Si] Sinclair, A.: Improved bounds for mixing rates of Markov chains and multicommodity flow. LFCS Report series178 (1991)
[SY] ShengLin Lu, Yau, H.T.: Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Commun. Math. Phys.156, 399–433 (1993)
[SZ] Stroock, D.W., Zegarlinski, B.: The logarithmic Sobolev inequality for discrete spin systems on a lattice. Commun. Math. Phys.149, 175–194 (1992)
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Communicated by M. Aizenman
Work partially supported by grant SCi-CT91-0695 of the Commission of European Communities
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Martinelli, F., Olivieri, E. Approach to equilibrium of Glauber dynamics in the one phase region. Commun.Math. Phys. 161, 447–486 (1994). https://doi.org/10.1007/BF02101929
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DOI: https://doi.org/10.1007/BF02101929