Diffusion in a sparsely connected space: A model for glassy relaxation

A model for diffusion in configuration space is proposed which combines the features of infinite dimensionality and low connectivity thought to be important for glassy relaxation. Specifically, a random walk amongst a set of N points, with each of the N(N-1)/2 pairs connected independently with probability p/N (and the mean connectivity p finite for N→∞), is considered. The model can be solved exactly by the replica method, but the behavior in the long-time regime is difficult to extract. From, instead, intuitive arguments based on the dominance for t→∞ of a particular type of statistical fluctuation in the network connectivity, the mean probability f(t) of return to the origin after time t is predicted to approach its infinite-time limit according to a ‘‘stretched-exponential’’ law, f(t)-f(∞)∼exp[-(t/τ${)}^{1/3}$] for all finite p, with τ∼‖p-1${\Vert }^{\mathrm{-}3}$ near the percolation threshold $p_c$=1.