The Annals of Probability

Towards a universality picture for the relaxation to equilibrium of kinetically constrained models

Fabio Martinelli and Cristina Toninelli

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Abstract

Recent years have seen a great deal of progress in our understanding of bootstrap percolation models, a particular class of monotone cellular automata. In the two-dimensional lattice $\mathbb{Z}^{2}$, there is now a quite satisfactory understanding of their evolution starting from a random initial condition, with a strikingly beautiful universality picture for their critical behavior. Much less is known for their nonmonotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM, each vertex is resampled (independently) at rate one by tossing a $p$-coin iff it can be infected in the next step by the bootstrap model. In particular, an infection can also heal, hence the nonmonotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own as they feature some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics. In this paper, we pave the way towards proving universality results for the characteristic time scales of KCM. Our novel and general approach gives the right tools to establish a close connection between the critical scaling of characteristic time scales for KCM and the scaling of the critical length in critical bootstrap models. When applied to the Fredrickson–Andersen $k$-facilitated models in dimension $d\ge2$, among the most studied KCM, and to the Gravner–Griffeath model, our results are close to optimal.

Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 324-361.

Dates
Received: January 2017
Revised: December 2017
First available in Project Euclid: 13 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1544691623

Digital Object Identifier
doi:10.1214/18-AOP1262

Mathematical Reviews number (MathSciNet)
MR3909971

Zentralblatt MATH identifier
07036339

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces

Keywords
Glauber dynamics kinetically constrained models spectral gap bootstrap percolation

Citation

Martinelli, Fabio; Toninelli, Cristina. Towards a universality picture for the relaxation to equilibrium of kinetically constrained models. Ann. Probab. 47 (2019), no. 1, 324--361. doi:10.1214/18-AOP1262. https://projecteuclid.org/euclid.aop/1544691623


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