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 $\beta$. Our analysis concerns the low-temperature regime $\beta \to \infty$, 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 $\beta \to \infty$ 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.

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References in corpus (4)

• Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States
• Low-temperature behavior of the multicomponent Widom-Rowlison model on finite square lattices
• Quasi-polynomial mixing of critical 2D random cluster models
• The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions