# zbMATH — the first resource for mathematics

Exponential convergence to equilibrium for the $$d$$-dimensional East model. (English) Zbl 1422.60160
Summary: Kinetically constrained models (KCMs) are interacting particle systems on $$\mathbb{Z}^d$$ with a continuous-time constrained Glauber dynamics, which were introduced by physicists to model the liquid-glass transition. One of the most well-known KCMs is the one-dimensional East model. Its generalization to higher dimension, the $$d$$-dimensional East model, is much less understood. Prior to this paper, convergence to equilibrium in the $$d$$-dimensional East model was proven to be at least stretched exponential, by P. Chleboun et al. [Ann. Fac. Sci. Toulouse, Math. (6) 24, No. 4, 717–743 (2015; Zbl 1333.60199)]. We show that the $$d$$-dimensional East model exhibits exponential convergence to equilibrium in all settings for which convergence is possible.
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text:
##### References:
 [1] Ludovic Berthier and Juan P. Garrahan, Numerical study of a fragile three-dimensional kinetically constrained model, The Journal of Physical Chemistry B 109 (2005), no. 8, 3578-3585. [2] Oriane Blondel, Front progression in the East model, Stochastic Processes and Their Applications 123 (2013), no. 9, 3430-3465. · Zbl 1291.60199 [3] Oriane Blondel, Nicoletta Cancrini, Fabio Martinelli, Cyril Roberto, and Cristina Toninelli, Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Processes and Related Fields 19 (2013), no. 3, 383-406. · Zbl 1321.82025 [4] Oriane Blondel, Aurélia Deshayes, and Cristina Toninelli, Front evolution of the Fredrickson-Andersen one spin facilitated model, Electronic Journal of Probability 24 (2019), 32. · Zbl 1406.60127 [5] Nicoletta Cancrini, Fabio Martinelli, Cyril Roberto, and Cristina Toninelli, Kinetically constrained spin models, Probability Theory and Related Fields 140 (2008), no. 3-4, 459-504. · Zbl 1176.82019 [6] Nicoletta Cancrini, Fabio Martinelli, Roberto H. Schonmann, and Cristina Toninelli, Facilitated oriented spin models: some non equilibrium results, Journal of Statistical Physics 138 (2010), no. 6, 1109-1123. · Zbl 1188.82048 [7] Paul Chleboun, Alessandra Faggionato, and Fabio Martinelli, Mixing time and local exponential ergodicity of the East-like process in $$\mathbb{Z} ^d$$, Annales de la faculté des Sciences de Toulouse 24 (2015), no. 4, 717-743. · Zbl 1333.60199 [8] Paul Chleboun, Alessandra Faggionato, and Fabio Martinelli, Relaxation to equilibrium of generalized East processes on $$\mathbb{Z} ^d$$: renormalization group analysis and energy-entropy competition, Annals of Probability 44 (2016), no. 3, 1817-1863. · Zbl 1343.60135 [9] Alessandra Faggionato, Fabio Martinelli, Cyril Roberto, and Cristina Toninelli, The East model: recent results and new progresses, Markov Processes and Related Fields 19 (2013), no. 3, 407-452. · Zbl 1321.60208 [10] Shirshendu Ganguly, Eyal Lubetzky, and Fabio Martinelli, Cutoff for the East process, Communications in Mathematical Physics 335 (2015), no. 3, 1287-1322. · Zbl 1408.82007 [11] Juan P. Garrahan, Peter Sollich, and Cristina Toninelli, Kinetically constrained models, Dynamical heterogeneities in glasses, colloids, and granular media, Oxford University Press, 2011. [12] Alice Guionnet and Boguslaw Zegarlinski, Lectures on logarithmic Sobolev inequalities, Séminaire de probabilités XXXVI, Springer, 2002, pp. 1-134. [13] Josef Jäckle and Siegfried Eisinger, A hierarchically constrained kinetic Ising model, Zeitschrift für Physik B Condensed Matter 84 (1991), no. 1, 115-124. [14] Sébastien Léonard, Peter Mayer, Peter Sollich, Ludovic Berthier, and Juan P. Garrahan, Non-equilibrium dynamics of spin facilitated glass models, Journal of Statistical Mechanics: Theory and Experiment (2007), 07017. [15] Thomas Mountford and Glauco Valle, Exponential convergence for the Fredrickson-Andersen one spin facilitated model, Journal of Theoretical Probability 32 (2019), no. 1, 282-302. · Zbl 1442.60103 [16] Felix Ritort and Peter Sollich, Glassy dynamics of kinetically constrained models, Advances in Physics 52 (2003), no. 4, 219-342. [17] Jan M. Swart, A course in interacting particle systems, arXiv:1703.10007v1 (2017).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.