# Exact asymptotics for Duarte and supercritical rooted kinetically constrained models

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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-02142008
Contributor : Cristina Toninelli <>
Submitted on : Tuesday, May 28, 2019 - 12:28:22 PM
Last modification on : Monday, August 3, 2020 - 3:39:54 AM

### Identifiers

• HAL Id : hal-02142008, version 1
• ARXIV : 1807.07519

### Citation

Laure Marêché, Fabio Martinelli, Cristina Toninelli. Exact asymptotics for Duarte and supercritical rooted kinetically constrained models. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), In press. ⟨hal-02142008⟩

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