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Exact asymptotics for Duarte and supercritical rooted kinetically constrained models. (English) Zbl 1451.60106
Summary: 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 [the last two authors, ibid. 47, 324–361 (2019; Zbl 07036339); the second author et al., Comm. Math. Phys. 369, 761–809 (2019; Zbl 1419.82037)], 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 [R. Morris, European J. Combin. 66 250–263 (2017; Zbl 1376.82090); the second author et al., loc. cit.] 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.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J27 Continuous-time Markov processes on discrete state spaces
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