Time Scale Separation and Dynamic Heterogeneity in the Low Temperature East Model

Chleboun, Paul; Faggionato, Alessandra; Martinelli, Fabio

doi:10.1007/s00220-014-1985-1

Cited by 25 publications

(67 citation statements)

References 38 publications

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“…The proof of this proposition, which is similar to that of [14,Proposition 3.4], is deferred to the Appendix.…”

Section: )mentioning

confidence: 89%

“…It is not difficult to see that in any dimension q c (Z d , U ) = 0. For d = 1, it was first proved in [1] that the relaxation time T rel (q) is finite for any q ∈ (0, 1], and it was later shown (see [1,10,14]) that it diverges as exp 1 + o (1) log(1/q) 2 2 log 2 as q ↓ 0. A similar scaling was later proved in any dimension d 1, see [15].…”

Section: Introductionmentioning

confidence: 99%

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Universality Results for Kinetically Constrained Spin Models in Two Dimensions

Martinelli

Morris

Toninelli

2018

Commun. Math. Phys.

105

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Kinetically constrained models (KCM) are reversible interacting particle systems on Z d with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as U-bootstrap percolation. KCM also display some of the peculiar features of the so-called "glassy dynamics", and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics.We consider two-dimensional KCM with update rule U, and focus on proving universality results for the mean infection time of the origin, in the same spirit as those recently established in the setting of U-bootstrap percolation. We first identify what we believe are the correct universality classes, which turn out to be different from those of U-bootstrap percolation. We then prove universal upper bounds on the mean infection time within each class, which we conjecture to be sharp up to logarithmic corrections. In certain cases, including all supercritical models, and the well-known Duarte model, our conjecture has recently been confirmed in [28]. In fact, in these cases our upper bound is sharp up to a constant factor in the exponent. For certain classes of update rules, it turns out that the infection time of the KCM diverges much faster than for the corresponding U-bootstrap process when the equilibrium density of infected sites goes to zero. This is due to the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the monotone bootstrap dynamics.

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“…The proof of this proposition, which is similar to that of [14,Proposition 3.4], is deferred to the Appendix.…”

Section: )mentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Universality Results for Kinetically Constrained Spin Models in Two Dimensions

Martinelli

Morris

Toninelli

2018

Commun. Math. Phys.

105

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“…Consider for example the one-dimensional East model [15] (and [13] for a review) for which a site can be updated iff its left neighbour is empty, namely U = {{− e 1 }}. As q ↓ 0, it holds E East µ (τ 0 ) = e (Θ(log q) 2 ) (1.2) and the scaling holds for T rel , see [1,8,9] where the sharp value of the constant has been determined. This divergence is much faster than for the corresponding Ubootstrap model, for which it holds T = Θ(1/q).…”

Section: Introductionmentioning

confidence: 99%

Exact asymptotics for Duarte and supercritical rooted kinetically constrained models

Marêché¹,

Martinelli²,

Toninelli³

2020

Ann. Probab.

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Kinetically constrained models (KCM) are reversible interacting particle systems on Z d with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as U-bootstrap percolation. Furthermore, KCM have an interest in their own since they display some of the most striking features of the liquid-glass transition, a major and longstanding open problem in condensed matter physics. A key issue for KCM is to identify the scaling of the characteristic time scales when the equilibrium density of empty sites, q, goes to zero. In [19,20] 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, the most studied critical 1-rooted model. We prove that the relaxation time and the mean infection time diverge for supercritical rooted KCM as e Θ((log q) 2 ) and for Duarte KCM as e Θ((log q) 4 /q 2 ) when q ↓ 0. These results prove the conjectures put forward in [20,22], and establish that the time scales for these KCM diverge much faster than for the corresponding 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.2010 Mathematics Subject Classification. Primary 60K35, secondary 60J27.

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“…Instead, there is not a direct connection with the cellular automata which allows to compute an upper bound on the time scales, and the best general upper bound is E µ (τ 0 ) ≤ exp(cL d c ) ( [9]). Though this bound has been refined for certain models leading in some cases the sharp behavior [9][10][11], the techniques are always ad hoc and valid only for very special choices of the constraints.…”

Section: Resultsmentioning

confidence: 99%

Interacting particle systems

et al. 2017

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Abstract. We present in this paper a number of recent and various works of statistical physics, which involve an interacting particle system. In kinetically constrained models, the particles are placed on Z d , with local constraints on the dynamics, that can slow down the evolution and reproduce the behaviour of glassy systems. The contact process (or Suspected-Infected-Susceptible) is a simple model for the spread of an infection on a (general) graph. In the Kuramoto model, the motion of the particles is determined by a diffusion system with mean-field interaction. Finally, in the modeling of the vertex reinforced jump process (a particle that interacts with itself via its path), we show an unexpected link with an interacting particle system. Résumé. Nous présentons dans ce papier plusieurs travaux récents et divers de physique statistique,qui mettent en jeu un système de particules en interaction. Dans les modèles cinétiquement contraints, les particules sont placées sur Z d avec des contraintes locales sur la dynamique qui peuvent ralentir celle-ci et reproduire le comportement des systèmes vitreux. Le processus de contact (ou SusceptibleInfected-Susceptible) modélise quant à lui la propagation d'une infection sur un graphe quelconque. Dans le modèle de Kuramoto, le mouvement des particules est déterminé par un système de diffusions, avec interaction de type champ moyen. Enfin, pour le processus de sauts renforcé par site, la particule interagit avec elle-même (avec sa trajectoire), la modélisation présentant un lien inattendu avec un système de particules en interaction.

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Time Scale Separation and Dynamic Heterogeneity in the Low Temperature East Model

Cited by 25 publications

References 38 publications

Universality Results for Kinetically Constrained Spin Models in Two Dimensions

Universality Results for Kinetically Constrained Spin Models in Two Dimensions

Exact asymptotics for Duarte and supercritical rooted kinetically constrained models

Interacting particle systems

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