Abstract
Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup P t . A fundamental and still largely open problem is the understanding of the long time behavior of δ η P t when the initial configuration η is sampled from a highly disordered state ν (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular b-ary tree , we study the above problem for the Ising and hard core gas (independent sets) models on . If ν is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove ν-almost sure weak convergence of δ η P t to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time t. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.
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References
Baxter, R.J.: Exactly solved models in statistical mechanics. Academic Press, London 1982
Berger, N., Kenyon, C., Mossel, E., Peres, Y.: Glauber dynamics on trees and hyperbolic graphs. Preprint, 2003
Bleher, P., Ruiz, J., Schonmann, R.H., Shlosman, S., Zagrebnov, V.: Rigidity of the critical phases on a Cayley tree. Moscow Math. J. 1, 345–363 (2001)
Bleher, P., Ruiz, J., Zagrebnov, V.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys. 79, 473–482 (1995)
Bodineau, T., Martinelli, F.: Some new results on the kinetic Ising model in a pure phase. J. Stat. Phys. 109 (1), 2002
Bray, A.J.: Theory of phase-ordering kinetics. Adv. Phys. 51 (1), 481–587 (2002)
Camia, F., De Santis, E., Newman, C.M.: Clusters and recurrence in the two-dimensional zero-temperature stochastic Ising model. Ann. Appl. Probab. 12 (2), 565–580 (2002)
Camia, F., Newman, C.M., Sidoravicius, V.: Approach to fixation for zero-temperature stochastic Ising models on the hexagonal lattice. In and out of equilibrium (Mambucaba, 2000), Progr. Probab. 51, Birkhäuser, 2002, pp. 163–183
Chayes, J.T., Chayes, L., Sethna, J.P., Thouless, D.J.: A mean field spin glass with short-range interactions. Commun. Math. Phys. 106, 41–89 (1986)
Evans, W., Kenyon, C., Peres, Y., Schulman, L.J.: Broadcasting on trees and the Ising model. Ann. Appl. Probability 10, 410–433 (2000)
Fontes, R., Schonmann, R., Sidoravicious, V.: Stretched exponential fixation in stochastic Ising models at zero temperature, Commum. Math. Phys. 228, 495–518 (2002)
Georgii, H.-O.: Gibbs measures and phase transitions. de Gruyter Studies in Mathematics 9, Walter de Gruyter & Co., Berlin, 1988
Howard, C.D., Newman, C.M.: The percolation transition for the zero-temperature stochastic Ising model on the hexagonal lattice. J. Statist. Phys. 111 (1–2), 57–72 (2003)
Howard, C.D.: Zero-temperature Ising spin dynamics on the homogeneous tree of degree three. J. Appl. Probab. 37 (3), 736–747 (2000)
Ioffe, D.: A note on the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37, 137–143 (1996)
Ioffe, D.: Extremality of the disordered state for the Ising model on general trees. Progress in Probability 40, 3–14 (1996)
Jonasson, J., Steif, J.E.: Amenability and phase transition in the Ising model. J. Theoretical Probability 12, 549–559 (1999)
Kelly, F.P.: Stochastic models of computer communication systems. J. Royal Stat. Soc. B 47, 379–395 (1985)
Ledoux, M.: The concentration of measure phenomenon. Am. Math. Soc., Providence, RI, 2001
Liggett, T.: Interacting particle systems. Springer-Verlag, New York, 1985
Liggett, T.: Stochastic interacting systems: contact, voter and exclusion processes. Springer, Berlin, 1999
Lyons, R.: Phase transitions on non amenable graphs. J. Math. Phys 41, 1099–1127 (2000)
Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Lecture notes in Mathematics 1717, Springer, Berlin, 1998, pp. 93–191
Martinelli, F., Sinclair, A., Weitz, D.: Glauber Dynamics on Trees: Boundary Conditions and Mixing Time. Commum. Math. Phys. 250, 301–334 (2004)
Martinelli, F., Sinclair, A., Weitz, D.: Fast mixing for independent sets, colorings and other models on trees. Submitted, 2004. Extended abstract appeared. In: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms, 2004, pp. 449–458
Mossel, E., Peres, Y.: Information flow on trees. Ann. Appl. Probability 13, 817–844 (2003)
Mossel, E.: Survey: information flow on trees. Graphs, morphisms and statistical physics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 63, Am. Math. Soc. Providence 2004, pp. 155–170
Peres, Y.: Probability on trees: an introductory climb. Lectures on probability theory and statistics (Saint-Flour, 1997), 1717, Lecture Notes in Mathematics, Springer, Berlin, 1999, pp. 193–280
Saloff-Coste, L.: Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour, 1996), Lecture notes in Mathematics 1665, Springer, Berlin, 1997, pp. 301–413
Schonmann, R.H., Shlosman, S.: Wulff droplets and the metastable relaxation of kinetic Ising models. Commum. Math. Phys. 194, 389–462 (1998)
Schonmann, R.H., Tanaka, N.I.: Lack of monotonicity in ferromagnetic Ising model phase diagrams. Ann. Appl. Probability 8, 234–245 (1998)
Spitzer, F.: Markov random fields on an infinite tree. Ann. Probability 3, 387–398 (1975)
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Caputo, P., Martinelli, F. Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree. Probab. Theory Relat. Fields 136, 37–80 (2006). https://doi.org/10.1007/s00440-005-0475-y
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DOI: https://doi.org/10.1007/s00440-005-0475-y