Facilitated spin models: recent and new results

TL;DR

This paper reviews recent mathematical results on kinetically constrained spin models, highlighting their glassy dynamics, analyzing long-term behavior, and introducing new extensions including interacting models and models on general graphs.

Contribution

It provides new rigorous analysis methods for KCSM, including interacting models and models on arbitrary graphs, advancing understanding of glassy dynamics.

Findings

Rigorous results on long-time behavior of KCSM

Analysis of interacting models with high-temperature Gibbs measures

Extension of models to general connected graphs

Abstract

Facilitated or kinetically constrained spin models (KCSM) are a class of interacting particle systems reversible w.r.t. to a simple product measure. Each dynamical variable (spin) is re-sampled from its equilibrium distribution only if the surrounding configuration fulfills a simple local constraint which \emph{does not involve} the chosen variable itself. Such simple models are quite popular in the glass community since they display some of the peculiar features of glassy dynamics, in particular they can undergo a dynamical arrest reminiscent of the liquid/glass transitiom. Due to the fact that the jumps rates of the Markov process can be zero, the whole analysis of the long time behavior becomes quite delicate and, until recently, KCSM have escaped a rigorous analysis with the notable exception of the East model. In these notes we will mainly review several recent mathematical results which, besides being applicable to a wide class of KCSM, have contributed to settle some debated questions arising in numerical simulations made by physicists. We will also provide some interesting new extensions. In particular we will show how to deal with interacting models reversible w.r.t. to a high temperature Gibbs measure and we will provide a detailed analysis of the so called one spin facilitated model on a general connected graph.