The influence of dimension on the relaxation process of East-like models: Rigorous results

We study facilitated models which extend to arbitrary dimensions the one-dimensional East process and which are supposed to catch some of the main features of the complex dynamics of fragile glasses. We focus on the low-temperature regime (small density c ≈ e−β of the facilitating sites). In the literature the relaxation process has been assumed to be quasi–one-dimensional and the equilibration time has been computed using the relaxation time of the East model (d = 1) on the equilibrium length scale Lc = (1/c)1/d in d-dimension. This led to a super-Arrhenius scaling for the relaxation time of the form Trel ≍ exp(β2/dlog2). In a companion paper, using renormalization group ideas and electrical networks methods, we rigorously establish that instead Trel ≍ exp(β2/2d log 2), contradicting the quasi–one-dimensional assumption. The above scaling confirms previous MCAMC simulations. Next we compute the relaxation time at finite and mesoscopic length scales, and show a dramatic dependence on the boundary conditions. Our final result is related to the out-of-equilibrium dynamics. Starting with a single facilitating site at the origin we show that, up to length scales L = O(Lc), its influence propagates much faster (on a logarithmic scale) along the diagonal direction than along the axes directions.