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Facilitated oriented spin models:some non equilibrium results

We analyze the relaxation to equilibrium for kinetically constrained spin models (KCSM) when the initial distribution $ν$ 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 $\bbZ$, for which the constraint requires that the East neighbor of the to-be-update vertex is vacant and the model on the binary tree introduced in \cite{Aldous:2002p1074}, for which the constraint requires the two children to be vacant. While the former model is ergodic at any $p\neq 1$, 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 $ν$ is concentrated on any configuration which does not contain a forever blocked site or if $ν$ is a Bernoulli($p&#39;$) product measure for any $p&#39;\neq 1$. For the model on the binary tree we prove similar results in the regime $p,p&#39;<p_c$ 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 $p&#39;>p_c$ and $p\leq p_c$ convergence to equilibrium cannot occur for all local functions. Finally we present a very simple argument (different from the one in \cite{Aldous:2002p1074}) based on a combination of combinatorial results and ``energy barrier&#39;&#39; considerations, which yields the sharp upper bound for the spectral gap of East when $p\uparrow 1$.