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Mixing time and local exponential ergodicity of the East-like process in d
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 4, pp. 717-743.

Le processus East est une chaîne linéaire de spins réversible et bien connue, qui représente le prototype d’une classe générale de systèmes de particules en interaction avec des contraintes modélisant des dynamiques vitreuses réelles. Dans ce papier, nous considérons une généralisation d-dimensionnelle du processus East et nous obtenons de nouveaux comportements hors équilibre. Bien que la convergence à l’équilibre réversible ne puisse pas avoir lieu en norme uniforme, de par la présence de configurations bloquantes, nous prouvons une ergodicité exponentielle locale pour les distributions initiales différentes de la probabilité stationnaire. Nous établissons également la croissance linéaire en la taille de la boîte du temps de mélange dans une boîte finie.

The East process, a well known reversible linear chain of spins, represents the prototype of a general class of interacting particle systems with constraints modeling the dynamics of real glasses. In this paper we consider a generalization of the East process living in the d-dimensional lattice and we establish new progresses on the out-of-equilibrium behavior. Despite the fact that convergence to the stationary reversible measure in the uniform norm cannot hold because of the presence of blocked configurations, we prove a form of (local) exponential ergodicity when the initial distribution is different from the stationary one. We also establish that the mixing time in a finite box grows linearly in the side of the box.

DOI : 10.5802/afst.1461
Paul Chleboun 1 ; Alessandra Faggionato 2 ; Fabio Martinelli 3

1 Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL.
2 Dipartimento di Matematica, Università La Sapienza, P.le Aldo Moro 2, 00185 Roma, Italy.
3 Dipartimento di Matematica e Fisica, Università Roma Tre, Largo S.L.Murialdo 00146, Roma, Italy.
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     title = {Mixing time and local exponential ergodicity of the {East-like} process in $\mathbb{Z}^d$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {717--743},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 24},
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Paul Chleboun; Alessandra Faggionato; 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 : Mathématiques, Série 6, Tome 24 (2015) no. 4, pp. 717-743. doi : 10.5802/afst.1461. https://afst.centre-mersenne.org/articles/10.5802/afst.1461/

[1] Aldous (D.) and Diaconis (P.).— Shuffling cards and stopping times, Amer. Math. Monthly 93, no.5, p.333-348 (1986) | MR | Zbl

[2] Aldous (D.) and Diaconis (P.).— The asymmetric one-dimensional constrained Ising model: rigorous results, J. Stat. Phys. 107, no. 5-6, p. 945-975 (2002). | MR | Zbl

[3] Bhatnagar (N.), Caputo (P.), Tetali (P.), and Vigoda (E.).— Analysis of top-swap shuffling for genome rearrangements, Ann. Appl. Probab. 17, no. 4, p. 1424-1445 (2007). | MR | Zbl

[4] Blondel (O.).— Front progression for the East model, Stochastic Process. Appl. 123, p. 3430-3465 (2013). | MR | Zbl

[5] Blondel (O.), Cancrini (N.), Martinelli (F.), Roberto (C.), and Toninelli (C.).— Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Process. Related Fields 19, p. 383-406 (2013). | MR

[6] Chleboun (P.) and Martinelli (F.).— Mixing time bounds for oriented kinetically constrained spin models, Electron. Commun. Probab. 18, no. 60, 9 (2013). | MR

[7] Cancrini (N.), Martinelli (F.), Roberto (C.), and Toninelli (C.).— Kinetically constrained spin models, Probab. Theory Rel. 140, no. 3-4, p. 459-504 (2008). | MR | Zbl

[8] Cancrini (N.), Martinelli (F.), Schonmann (R.), and Toninelli (C.).— Facilitated oriented spin models: some non equilibrium results, J. Stat. Phys. 138, no. 6, p. 1109-1123 (2010). | MR | Zbl

[9] Caputo (P.), Lubetzky (E.), Martinelli (F.), Sly (A.), and F. Toninelli (L.).— Dynamics of 2+1 dimensional SOS surfaces above a wall: slow mixing induced by entropic repulsion, Annals of Probability 42, no. 4, p. 1516-1589 (2012). | MR

[10] Chleboun (P.), Faggionato (A.), and Martinelli (F.).— Time scale separation and dynamic heterogeneity in the low temperature East model, Commun. Math. Phys. 328, p. 955-993 (2014). | MR | Zbl

[11] Chleboun (P.), Faggionato (A.), and Martinelli (F.).— Time scale separation in the low temperature East model: rigorous results, Journal of Statistical Mechanics: Theory and Experiment 2013, no. 04, L04001 (2013).

[12] Chleboun (P.), Faggionato (A.), and Martinelli (F.).— Relaxation to equilibrium of generalized East processes on Z d : Renormalization group analysis and energy-entropy competition, Annals of Probability, to appear (2014). | MR

[13] Chleboun (P.), Faggionato (A.), and Martinelli (F.).— The influence of dimension on the relaxation process of East-like models, European Physics Letters 107, no. 3 (2014).

[14] F. Chung, Diaconis (P.), and Graham (R.).— Combinatorics for the East model, Adv. in Appl. Math. 27, no. 1, p. 192-206 (2001). | MR | Zbl

[15] Diaconis (P.).— The cutoff phenomenon in finite Markov chains, Proc. Nat. Acad. Sci. U.S.A. 93, no. 4, p. 1659-1664 (1996). | MR | Zbl

[16] Diaconis (P.) and Shahshahani (M.).— Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57, no. 2, p. 159-179 (1981). | MR | Zbl

[17] Faggionato (A.), Martinelli (F.), Roberto (C.), and Toninelli (C.).— The East model: recent results and new progresses, Markov Processes and Related Fields 19, p. 407-458 (2013). | MR

[18] Faggionato (A.), Martinelli (F.), Roberto (C.), and Toninelli (C.).— Aging through hierarchical coalescence in the East model, Commun. Math. Phys. 309, p. 459-495 (2012). | MR | Zbl

[19] Faggionato (A.), Martinelli (F.), Roberto (C.), and Toninelli (C.).— Universality in one dimensional hierarchical coalescence processes, Ann. Probab. 40, no. 4, p. 1377-1435 (2012). | MR | Zbl

[20] Faggionato (A.), Roberto (C.), and Toninelli (C.).— Universality for one-dimensional hierarchical coalescence processes with double and triple merges, Annals of Applied Probability 24, no. 2, p. 476-525 (2014). | MR

[21] Ganguly (S.), Lubetzky (E.), and Martinelli (F.).— Cutoff for the East process, Comm. Math. Phys. Comm. Math. Phys. 335, no. 3, p. 1287-1322 (2015). | MR

[22] Elmatad (Y.S.), Chandler (D.), and Garrahan (J. P.).— Corresponding States of Structural Glass Formers, J. Phys. Chem. B 113, p. 5563-5567 (2009).

[23] Keys (A. S.), Hedges (L.), Garrahan (J. P.), Glotzer (S. C.), and Chandler (D.).— Excitations are localized and relaxation is hierarchical in glass-forming liquids, Phys. Rev. X 1, 021013 (2011).

[24] Garrahan (J. P.) and Chandler (D.).— Coarse-grained microscopic model of glass formers, Proc. Nat. Acad. Sci. U.S.A. 100, no. 17, p. 9710-9714 (2003).

[25] Garrahan (J. P.), Sollich (P.), and Toninelli (C.).— Kinetically constrained models, in ÓDynamical heterogeneities in glasses, colloids, and granular mediaÓ, Oxford Univ. Press, Eds.: L. Berthier, G. Biroli, J-P Bouchaud, L. Cipelletti and W. van Saarloos (2011).

[26] Jäckle (J.) and Eisinger (S.).— A hierarchically constrained kinetic Ising model, Z. Phys. B: Condens. Matter 84, no. 1, p. 115-124 (1991).

[27] Harris (T. E.).— Nearest-neighbor Markov interaction processes on multidimensional lattices, Advances in Mathematics 9, no. 1, p. 66-89 (1972). | MR | Zbl

[28] Keys (A. S.), Garrahan (J. P.), and Chandler (D.).— Calorimetric glass transition explained by hierarchical dynamic facilitation, Proc. Nat. Acad. Sci. U.S.A. 110, no. 12, p. 4482-4487 (2013).

[29] Levin (D. A.), Peres (Y.), and Wilmer (E. L.).— Markov chains and mixing times, American Mathematical Society (2008). | MR | Zbl

[30] Liggett (T. M.).— Interacting particle systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276, Springer-Verlag, New York (1985). | MR | Zbl

[31] Liggett (T. M.).— Stochastic interacting systems: contact, voter and exclusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 324, Springer-Verlag, Berlin (1999). | MR | Zbl

[32] Peres (Y.) and Sly (A.).— Mixing of the upper triangular matrix walk, Probab. Theory Rel. 156, no. 3-4, p. 581-591 (2012). | MR | Zbl

[33] Ritort (F.) and Sollich (P.).— Glassy dynamics of kinetically constrained models, Advances in Physics 52, no. 4, p. 219-342 (2003).

[34] Saloff-Coste (L.).— Lectures on finite Markov chains (P. Bernard, ed.), Lecture Notes in Mathematics, vol. 1665, Springer Berlin Heidelberg (1997). | MR | Zbl

[35] Sollich (P.) and Evans (M. R.).— Glassy time-scale divergence and anomalous coarsening in a kinetically constrained spin chain, Phys. Rev. Lett 83, p. 3238-3241 (1999).

[36] Sollich (P.) and Evans (M. R.).— Glassy dynamics in the asymmetrically constrained kinetic Ising chain, Phys. Rev. E (2003), 031504.

[37] Toninelli (C.) and Biroli (G.).— A new class of cellular automata with a discontinuous glass transition, J. Stat. Phys. 130, no. 1, p. 83-112 (2008). | MR | Zbl

[38] Valiant (P.).— Linear bounds on the North-East model and higher-dimensional analogs, Advances in Applied Mathematics 33, no. 1, p. 40-50 (2004). | MR | Zbl

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