×

Bound on the mass gap for finite volume stochastic Ising models at low temperature. (English) Zbl 0679.60102

Summary: We consider a sequence of finite volume \(\Lambda \subset \mathbb Z^ d\), \(d\geq 2\), reversible stochastic Ising models in the low temperature regime and having invariant measures satisfying free boundary conditions. We show that associated with the models are random hitting times whose expectations, regarded as a function of \(\Lambda\), grow exponentially in \(| \Lambda |^{(d-1)/d}\); moreover, the mass gaps for the models shrink exponentially fast in \(| \Lambda |^{(d-1)/d}\). A geometrical lemma is employed in the analysis which states that if a Peierls’ contour is sufficiently small relative to the faces of \(\Lambda\), then the fraction of the contour tangent to the faces is less than a constant smaller than one.

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Liggett, T.M.: The stochastic evolution of infinite systems of interacting particles. Lecture Notes in Mathematics, vol.598. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0363.60109
[2] Liggett, T.M.: Interacting particle systems. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0559.60078
[3] Lamperti, J.: Stochastic processes, a survey of mathematical theory. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0365.60001
[4] Holley, R.A., Stroock, D.: Applications of the stochastic Ising model to the Gibbs states. Commun. Math. Phys.48, 249–265 (1976) · doi:10.1007/BF01617873
[5] Sullivan, W.G.: A unified existence and ergodic theorem for Markov evolution of random fields. Z. Wahrsheinlichkeitstheorie Verw Geb.31, 47–56 (1974) · Zbl 0295.60051 · doi:10.1007/BF00538715
[6] Dobrushin, R.L.: Markov processes with a large number of locally interacting components, existence of the limiting process and its ergodicity. Probl. Peredaci Inform.7, 70–87 (1971)
[7] Thomas, L.E., Yin, Z.: Approach to equilibrium for random walks on graphs and for stochastic infinite particle systems. J. Math. Phys.27, 2475–2477 (1986) · Zbl 0604.60100 · doi:10.1063/1.527310
[8] Sokal, A.D., Thomas, L.E.: Absence of mass gap for a class of stochastic contour models. J. Stat. Phys.51, 907–947 (1988) · Zbl 1086.60522 · doi:10.1007/BF01014892
[9] Sokal, A.D., Thomas, L.E.: Exponential convergence to equilibrium for a class of random walk models. J. Stat. Phys.54, 797–828 (1989) · Zbl 0668.60061 · doi:10.1007/BF01019776
[10] Schonmann, R.H.: Second order large deviation estimates for ferromagnetic systems in the phase coexistence region. Commun. Math. Phys.112, 409–422 (1987) · doi:10.1007/BF01218484
[11] Ruelle, D.: Statistical mechanics, rigorous results. New York: W. A Benjamin 1969 · Zbl 0177.57301
[12] Feller, W.: An Introduction to Probability theory and its Applications I, New York: Wiley 1968, Chap. XV of vol. I and p. 491–495 of vol. II · Zbl 0155.23101
[13] Nummelin, E.: General Irreducible Markov chains and Non-negative Operators. New York: Cambridge University Press 1984, Chap. 5 · Zbl 0551.60066
[14] Dym, H., McKean, H.P.: Fourier Series and Integrals. New York: Academic Press 1972 · Zbl 0242.42001
[15] Chayes, J., Chayes, L., Schonmann, R.: Exponential decay of connectivities in the two dimensional Ising model. J. Stat. Phys.49, 433–445 (1987) · Zbl 0962.82522 · doi:10.1007/BF01009344
[16] Capocaccia, M., Cassandro, M., Olivieri, E.: A study of metastability in the Ising model. Commun. Math. Phys.39, 185–205 (1974) · doi:10.1007/BF01614240
[17] Lebowitz, J., Schonmann, R.: On the asymptotics of occurrence times of rare events in stochastic spin systems. J. Stat. Phys.48, 727–751 (1987) · Zbl 1084.82521 · doi:10.1007/BF01019694
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.