Long time behaviour of one facilitated kinetically constrained models: results and open problems

TL;DR

This paper reviews the long-term behavior of kinetically constrained models (KCMs), focusing on open problems and recent partial results in their non-stationary dynamics, especially at low equilibrium densities.

Contribution

It discusses open questions, presents partial new results, and conjectures regarding the non-stationary evolution of facilitated KCMs and related processes.

Findings

Progress in understanding stationary evolution of KCMs

Identification of open problems in low-density regimes

Partial results and conjectures on non-stationary dynamics

Abstract

Kinetically constrained models (KCMs) are interacting particle systems introduced in the '80s by physicists to have accessible stochastic models with glassy-type dynamics. The key mechanism behind the complex evolution of these otherwise simple models is the so-called dynamical facilitation, a feature embedded into the models via appropriate kinetic constraints. KCMs are reversible with respect to a Bernoulli product measure, and the analysis of their stationary evolution has witnessed significant progress in the last decade. Unfortunately, in the interesting regime when the equilibrium density of the facilitating vertices is small, many fundamental questions concerning the non-stationary evolution of even the simplest models remain unsolved. In this paper, we discuss some of these questions, along with partial new results and conjectures, for the one facilitated model and its variants, as well as for the biased annihilating branching process.