Facilitated Oriented Spin Models: Some Non Equilibrium Results

We perform the rigorous analysis of the relaxation to equilibrium for some facilitated or kinetically constrained spin models (KCSM) when the initial distribution ! is different from the reversible one, μ. This setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase. We concentrate on two basic oriented KCSM: the East model on Z, for which the constraint requires that the East neighbor of the to-be-update vertex is vacant and the AD model on the binary tree introduced in Aldous and Diaconis (J. Stat. Phys. 107(5–6):945– 975, 2002), for which the constraint requires the two children to be vacant. It is important to observe that, while the former model is ergodic at any p != 1, the latter displays an ergodicity breaking transition at pc = 1/2. For the East we prove exponential convergence to equilibrium with rate depending on the spectral gap if ! is concentrated on any configuration which does not contain a forever blocked site or if ! is a Bernoulli(p") product measure for any p" != 1. For the model on the binary tree we prove similar results in the regime p,p" < pc and under the (plausible) assumption that the spectral gap is positive for p < pc. By constructing a proper test function, we also prove that if p" > pc and p # pc convergence to equilibrium cannot occur for all local functions. Finally, in a short appendix, we present a very simple argument, different from the one given in Aldous and Diaconis (J. Stat. Phys. 107(5–6):945–975, 2002), based on a combination of some combinatorial results together with “energy barrier” considerations, which yields the sharp upper bound for the spectral gap of East when p $ 1.