Tunneling behavior of Ising and Potts models in the low-temperature regime

Abstract

We consider the ferromagnetic q-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature β. Our analysis concerns the low-temperature regime β → ∞, in which this multi-spin system has q stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the q-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria as β → ∞ in probability, in expectation, and in distribution and obtain tight bounds on the mixing time as side-result. In the special case q = 2, our results characterize the tunneling behavior of the Ising model on grid graphs.

Critical Configurations and Tube of Typical Trajectories for the Potts and Ising Models with Zero External Field

“…Set of all the cycles and extended cycles of X M(A) Set of configurations with all spins s, except those, which are r , in a rectangle a ×b with a bar 1×l adjacent to one of the sides of length b, with 1 ≤ l ≤ b−1 H (r, s)B 1 1,K (r , s) ∪ K −2 h=2B h 1,K −1 (r , s) H (r, s)B 1 1,K (r , s) ∪ K −2 h=2B h 1,K −1 (r , s) Q(r, s)R 2,K −1 (r , s) ∪ K −2 h=2B h 1,K (r , s) Q(r, s)R 2,K −1 (r , s) ∪ K −2 h=2B h 1,K (r , s) P(r, s)B K −1 1,K (r , s) P(r, s)B K −1 1,K (r , s) W (h) j (r, s)B h j,K (r , s) =B K −h L− j−1,K (r , s) for j = 2, . .…”

“…Related work. Our work concludes the study of the metastability of the Potts model in the low-temperature regime first initiated in [1], where the authors derive the asymptotic behavior of the first hitting time associated with the transitions (b) and (c) above. They obtain convergence results in probability, in expectation and in distribution.…”

“…While there are some overlaps between the two papers, the works of the two groups were carried out independently of each other. They find sharp estimates for the tunneling time (the so-called prefactor) using the potential-theoretic approach and extending the results in [1]. In the second part of their manuscript, the authors focus on the two-dimensional setting.…”

Metastable Markov chains

“…The mean-field Potts model is another spin dynamics in which the spin may take more than two values. It has been examine recently in [92] and by Nardi and Zocca in [103].…”

Condensation of Non-reversible Zero-Range Processes

“…Metastability is a generic phenomenon that occurs in several of models in probability theory and statistical physics, such as random perturbations of dynamical systems [9,14,25], low-temperature ferromagnetic spin systems [6,11,12,28,29], the stochastic partial differential equations [5], and the system of sticky particles [4,7,18,22]. For an extensive discussion of recent developments in this field, certain monographs [10,30] can be referred to.…”