## Electronic Communications in Probability

### Exponential convergence to equilibrium for the $d$-dimensional East model

Laure Marêché

#### Abstract

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 Chleboun, Faggionato and Martinelli in 2015. We show that the $d$-dimensional East model exhibits exponential convergence to equilibrium in all settings for which convergence is possible.

#### Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 55, 10 pp.

Dates
Received: 2 May 2019
Accepted: 19 August 2019
First available in Project Euclid: 13 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1568361881

Digital Object Identifier
doi:10.1214/19-ECP261

Mathematical Reviews number (MathSciNet)
MR4003129

Zentralblatt MATH identifier
1422.60160

#### Citation

Marêché, Laure. Exponential convergence to equilibrium for the $d$-dimensional East model. Electron. Commun. Probab. 24 (2019), paper no. 55, 10 pp. doi:10.1214/19-ECP261. https://projecteuclid.org/euclid.ecp/1568361881

#### 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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [10] Shirshendu Ganguly, Eyal Lubetzky, and Fabio Martinelli, Cutoff for the East process, Communications in Mathematical Physics 335 (2015), no. 3, 1287–1322.
• [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.
• [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).