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January 2020 Exact asymptotics for Duarte and supercritical rooted kinetically constrained models
Laure Marêché, Fabio Martinelli, Cristina Toninelli
Ann. Probab. 48(1): 317-342 (January 2020). DOI: 10.1214/19-AOP1362

Abstract

Kinetically constrained models (KCM) are a class of interacting particle systems which represent a natural stochastic (and nonmonotone) counterpart of the family of cellular automata known as $\mathcal{U}$-bootstrap percolation. A key issue for KCM is to identify the divergence of the characteristic time scales when the equilibrium density of empty sites, $q$, goes to zero. In (Ann. Probab. 47 (2019) 324–361; Comm. Math. Phys. 369 (2019) 761–809), 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. 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 (European J. Combin. 66 (2017) 250–263; Comm. Math. Phys. 369 (2019) 761–809) for these models, 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.

Citation

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Laure Marêché. Fabio Martinelli. Cristina Toninelli. "Exact asymptotics for Duarte and supercritical rooted kinetically constrained models." Ann. Probab. 48 (1) 317 - 342, January 2020. https://doi.org/10.1214/19-AOP1362

Information

Received: 1 July 2018; Published: January 2020
First available in Project Euclid: 25 March 2020

zbMATH: 07206760
MathSciNet: MR4079438
Digital Object Identifier: 10.1214/19-AOP1362

Subjects:
Primary: 60K35
Secondary: 60J27

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 1 • January 2020
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