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Kinetically constrained spin models
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  • Published: 22 May 2007

Kinetically constrained spin models

  • N. Cancrini1,
  • F. Martinelli2,
  • C. Roberto3 &
  • …
  • C. Toninelli4 

Probability Theory and Related Fields volume 140, pages 459–504 (2008)Cite this article

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Abstract

We analyze the density and size dependence of the relaxation time for kinetically constrained spin models (KCSM) intensively studied in the physics literature as simple models sharing some of the features of the glass transition. KCSM are interacting particle systems on \({\mathbb{Z}}^d\) with Glauber-like dynamics, reversible w.r.t. a simple product i.i.d Bernoulli(p) measure. The essential feature of a KCSM is that the creation/destruction of a particle at a given site can occur only if the current configuration around it satisfies certain constraints which completely define each specific model. No other interaction is present in the model. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density p remained open for most KCSM (with the notably exception of the East model in d = 1; Aldous and Diaconis in J Stat Phys 107(5–6):945–975, 2002). Here for the first time we: (i) identify the ergodicity region by establishing a connection with an associated bootstrap percolation model; (ii) develop a novel multi-scale approach which proves positivity of the spectral gap in the whole ergodic region; (iii) establish, sometimes optimal, bounds on the behavior of the spectral gap near the boundary of the ergodicity region and (iv) establish pure exponential decay at equilibrium for the persistence function, i.e. the probability that the occupation variable at the origin does not change before time t. Our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations.

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Author information

Authors and Affiliations

  1. Dip. Matematica Univ.l’Aquila, 1-67100, L’Aquila, Italy

    N. Cancrini

  2. Dip. Matematica, Univ. Roma Tre, Largo S.L.Murialdo, 00146, Roma, Italy

    F. Martinelli

  3. L.A.M.A., Univ Marne-la-Vallée, 5 bd Descartes, 77454, Marne-la-Vallée, France

    C. Roberto

  4. Laboratoire de Probabilités et Modèles, Alèatoires CNRS-UMR 7599 Universités, Paris VI-VII 4, Place Jussieu, 75252, Paris Cedex 05, France

    C. Toninelli

Authors
  1. N. Cancrini
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  2. F. Martinelli
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  3. C. Roberto
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  4. C. Toninelli
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Corresponding author

Correspondence to F. Martinelli.

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Cancrini, N., Martinelli, F., Roberto, C. et al. Kinetically constrained spin models. Probab. Theory Relat. Fields 140, 459–504 (2008). https://doi.org/10.1007/s00440-007-0072-3

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  • Received: 10 October 2006

  • Revised: 14 March 2007

  • Published: 22 May 2007

  • Issue Date: March 2008

  • DOI: https://doi.org/10.1007/s00440-007-0072-3

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Keywords

  • Glauber dynamics
  • Spectral gap
  • Constrained models
  • Dynamical phase transition
  • Glass transition

Mathematics Subject Classification (2000)

  • 60K35
  • 60K40
  • 82B20
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