Title: Relaxation and mixing of kinetically constrained models.

Abstract: We study the relaxation and out-of-equilibrium dynamics of a family of kinetically constrained models (KCMs) called the d-dimensional East-like processes. KCMs are spin systems on integer lattices, where each vertex is labelled either 0 or 1, which evolve according to a very simple rule: i) with rate one and independently for each vertex, a new value 1/0 is proposed with probability 1-q and q respectively; ii) the proposed value is accepted if and only if the neighbouring spins satisfy a certain constraint. Despite of their apparent simplicity, KCMs pose very challenging and interesting problems due to the hardness of the constraints and lack of monotonicity. The out-of-equilibrium dynamics are extremely rich and display many of the key features of the dynamics of real glasses, such as; an ergodicity breaking transition at some critical value, huge relaxation times close to the critical point, and dynamic heterogeneity (non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium). We discuss recent advances on the out-of-equilibrium dynamics of the East-like processes, including the dependence of the relaxation and mixing time on the system size, density, and dimension. We also look at simulations which motivate some interesting limit shape conjectures.
This is joint work with Alessandra Faggionato and Fabio Martinelli.