Universality for critical KCM: infinite number of stable directions
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Universality for critical KCM: infinite number of stable directions Ivailo Hartarsky1,2
· Laure Marêché3 · Cristina Toninelli2
Received: 29 July 2019 / Revised: 9 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Kinetically constrained models (KCM) are reversible interacting particle systems on Zd with continuous-time constrained Glauber dynamics. They are a natural nonmonotone stochastic version of the family of cellular automata with random initial state known as U -bootstrap percolation. KCM have an interest in their own right, owing to their use for modelling the liquid-glass transition in condensed matter physics. In two dimensions there are three classes of models with qualitatively different scaling of the infection time of the origin as the density of infected sites vanishes. Here we study in full generality the class termed ‘critical’. Together with the companion paper by Hartarsky et al. (Universality for critical KCM: finite number of stable directions. arXiv e-prints arXiv:1910.06782, 2019) we establish the universality classes of critical KCM and determine within each class the critical exponent of the infection time as well as of the spectral gap. In this work we prove that for critical models with an infinite number of stable directions this exponent is twice the one of their bootstrap percolation counterpart. This is due to the occurrence of ‘energy barriers’, which determine the dominant behaviour for these KCM but which do not matter for the monotone bootstrap dynamics. Our result confirms the conjecture of Martinelli et al. (Commun Math Phys 369(2):761–809. https://doi.org/10.1007/s00220-018-3280-z, 2019), who proved a matching upper bound. Keywords Kinetically constrained models · Bootstrap percolation · Universality · Glauber dynamics · Spectral gap Mathematics Subject Classification Primary 60K35; Secondary 82C22 · 60J27 · 60C05
This work was supported by European Research Council Starting Grant 680275 MALIG and by Agence Nationale de la Recherche-15-CE40-0020-01. Extended author information available on the last page of the article
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1 Introduction Kinetically constrained models (KCM) are interacting particle systems on the integer lattice Zd , which were introduced in the physics literature in the 1980s by Fredrickson and Andersen [16] in order to model the liquid-glass transition (see e.g. [17,31] for reviews), a major and still largely open problem in condensed matter physics [5]. A generic KCM is a continuous-time Markov process of Glauber type characterised by a finite collection U of finite nonempty subsets of Zd \{0}, its update family. A configuration ω is defined by assigning to each site x ∈ Zd an occupation variable ωx ∈ {0, 1}, corresponding to an empty or occupied site respectively. Each site x ∈ Zd waits an independent, mean one, exponential time and then, iff there exists U ∈ U such that ω y = 0 for all y ∈ U + x, site x is updated to empty with probability q and to occupied with probability 1 − q. Since each U ∈ U i
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