Facilitated Spin Models: Recent and New Results

  • N. CancriniEmail author
  • F. MartinelliEmail author
  • C. Robert
  • C. Toninelli
Part of the Lecture Notes in Mathematics book series (LNM, volume 1970)


Facilitated or kinetically constrained spin models (KCSM) are a class of interacting particle systems reversible w.r.t. to a simple product measure. Each dynamical variable (spin) is re-sampled from its equilibrium distribution only if the surrounding configuration fulfills a simple local constraint which does not involve the chosen variable itself. Such simple models are quite popular in the glass community since they display some of the peculiar features of glassy dynamics, in particular they can undergo a dynamical arrest reminiscent of the liquid/glass transition. Due to the fact that the jumps rates of the Markov process can be zero, the whole analysis of the long time behavior becomes quite delicate and, until recently, KCSM have escaped a rigorous analysis with the notable exception of the East model. In these notes we will mainly review several recent mathematical results which, besides being applicable to a wide class of KCSM, have contributed to settle some debated questions arising in numerical simulations made by physicists. We will also provide some interesting new extensions. In particular we will show how to deal with interacting models reversible w.r.t. to a high temperature Gibbs measure and we will provide a detailed analysis of the so called one spin facilitated model on a general connected graph.


Gibbs Measure Spin Model Dirichlet Form Interact Particle System Glauber Dynamic 
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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1. Dip. Matematica Univl’AquilaItaly
  2. 2.Dip. MatematicaUniv. Roma Tre, Largo S.L. MurialdoPragueCzech Republic
  3. 3. Universite Paris-est, L.A.M.A. UMR 8050France
  4. 4. Laboratoire de Probabilit´es et Mod`eles Al`eatoires CNRS-UMR 7599 12Universit´es Paris VI-VII 4, Place Jussieu F-75252 Paris Cedex 05France

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