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 nu is different from the reversible one, mu. 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 a"currency sign, 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 not equal 1, the latter displays an ergodicity breaking transition at p (c) =1/2. For the East we prove exponential convergence to equilibrium with rate depending on the spectral gap if nu is concentrated on any configuration which does not contain a forever blocked site or if nu is a Bernoulli(p') product measure for any p'not equal 1. For the model on the binary tree we prove similar results in the regime p,p'< p (c) and under the (plausible) assumption that the spectral gap is positive for p < p (c) . By constructing a proper test function, we also prove that if p'> p (c) and pa parts per thousand currency signp (c) 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 pa dagger 1.