Mathematics Colloquia and Seminars

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Description

Kinetically constrained spin models are lattice 0-1 spins evolving under a Glauber (or Metropolis) type of dynamics, usually reversible w.r.t. trivial Bernoulli product measure, in which a single updating at a given vertex can occur only if the current configuration around that vertex satisfies certain specific constraints. They are intensively studied in the physics literature in connection with "glass and jamming transitions" and are closely related to bootstrap percolation models. The simplest example is the so called Fredrickon-Andersen (FA-f) in which the spin at site x can flip only if "f" among its neighbors have value zero. Due to the degenaracy of the jump rates the configuration space can be broken into different irreducible components and the invariant measure is not unique. Such non uniqueness may lead to dynamical phase transition in the thermodynamic limit. The only rigorous result available so far has been obtained some years ago by Aldous and Diaconis for the East model in one dimension. In this talk I will report on a series of new results for a wide class of models in dimension greater than one obtained in collaboration with N. Cancrini (Rome), C. Roberto and C. Toninelli (Paris). The main achievements are upper bounds on the relaxation time up to the critical point, exponential decay of the so called "persistence function" and sharp asympotics near the critical point. Some of our findings contradict some previous conjectures based on numerical simulations. Our technique is based on a novel block dynamics which takes into account the kinetical constraints on larger and larger scales.