2018/07/19 by Laure Marêché, Fabio Martinelli, Marêché, Laure +3
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1807.07519
arxiv created 2018/07/19 · arxiv updated 2018/07/20
Kinetically constrained models (KCM) are reversible interacting particle systems on \mathbb Zd 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 eΘ((log q)2) and for Duarte KCM as eΘ((log q)4/q2) 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.