Exact asymptotics for Duarte and supercritical rooted kinetically constrained models

TL;DR

This paper establishes precise asymptotic lower bounds for the divergence of relaxation and infection times in certain kinetically constrained models as the density of empty sites approaches zero, confirming longstanding conjectures.

Contribution

It introduces a novel technique to prove matching lower bounds for the divergence of time scales in supercritical rooted KCM and Duarte models, advancing understanding of their dynamics.

Findings

Relaxation time diverges as e^{Θ((log q)^2)} for supercritical rooted KCM.

Duarte KCM's time diverges as e^{Θ((log q)^4/q^2)} as q approaches zero.

Results confirm previous conjectures and highlight the impact of energy barriers on KCM dynamics.

Abstract

Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as $\mathcal U$-bootstrap percolation. Furthermore, KCM have an interest in their own since they display some of the most striking features of the liquid-glass transition, a major and longstanding open problem in condensed matter physics. A key issue for KCM is to identify the scaling of the characteristic time scales when the equilibrium density of empty sites, $q$, goes to zero. In [19,20] a general scheme was devised to determine a sharp upper bound for these time scales. Our paper is devoted to developing a (very different) technique which allows to prove matching lower bounds. We analyse the class of two-dimensional supercritical rooted KCM and the Duarte KCM, the most studied critical $1$-rooted model. We prove that the relaxation time and the mean infection time diverge for supercritical rooted KCM as $e^{\Theta((\log q)^2)}$ and for Duarte KCM as $e^{\Theta((\log q)^4/q^2)}$ when $q\downarrow 0$. These results prove the conjectures put forward in [20,22], and establish that the time scales for these KCM diverge much faster than for the corresponding $\mathcal U$-bootstrap processes, the main reason being the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the bootstrap dynamics.