Stochastic Ising model with flipping sets of spins and fast decreasing temperature

TL;DR

This paper studies a generalized stochastic Ising model with temperature decreasing to zero, analyzing how often individual spins flip under various conditions, and classifies the dynamics into finite, infinite, or mixed flip behaviors.

Contribution

It introduces a generalized Glauber dynamics with simultaneous spin flips on periodic graphs and provides conditions for different flip behavior classifications.

Findings

Conditions for spins flipping finitely or infinitely often are established.

Results apply to zero-temperature and cubic lattice cases.

Examples across various dimensions and graph types are provided.

Abstract

This paper deals with the stochastic Ising model with a temperature shrinking to zero as time goes to infinity. A generalization of the Glauber dynamics is considered, on the basis of the existence of simultaneous flips of some spins. Such dynamics act on a wide class of graphs which are periodic and embedded in $\mathbb{R}^d$. The interactions between couples of spins are assumed to be quenched i.i.d. random variables following a Bernoulli distribution with support $\{-1,+1\}$. The specific problem here analyzed concerns the assessment of how often (finitely or infinitely many times, almost surely) a given spin flips. Adopting the classification proposed in \cite{GNS}, we present conditions in order to have models of type $\mathcal{F}$ (any spin flips finitely many times), $\mathcal{I}$ (any spin flips infinitely many times) and $\mathcal{M}$ (a mixed case). Several examples are provided in all dimensions and for different cases of graphs. The most part of the obtained results holds true for the case of zero-temperature and some of them for the cubic lattice $\mathbb{L}_d=(\mathbb{Z}^d, \mathbb{E}_d)$ as well.