Ising Model on the Torus and PCA Dynamics: Reversibility, Irreversibility, and Fast Tunneling

Abstract

In this chapter, we present a class of PCA (Probabilistic Cellular Automata) that can be used for approximate sampling the Gibbs measure. We list a series of results about them, restricting the discussion to the nearest-neighbor Ising model. For both the weakly and strongly coupled spins, we show how it is possible to explicitly evaluate the accuracy of our approximation scheme. Moreover, in the strong coupling regime (low temperature), we show how our procedure may drastically improve the known results about the convergence of the system to the stationary distribution. An important ingredient in this context is the use of an irreversible dynamics, which let new interesting states (the so-called Ising waves) arise.