Escaping from cycles through a glass transition

Abstract

A random walk is performed over a disordered media composed of N sites random and uniformly distributed inside a d-dimensional hypercube. The walker cannot remain in the same site and hops to one of its n neighboring sites with a transition probability that depends on the distance D between sites according to a cost function E(D). The stochasticity level is parametrized by a formal temperature T. In the case T=0, the walk is deterministic and ergodicity is broken: the phase space is divided in a O(N) number of attractor basins of two-cycles that trap the walker. For d=1, analytic results indicate the existence of a glass transition at T1=1/2 as N->∞. Below T1, the average trapping time in two-cycles diverges and an out-of-equilibrium behavior appears. Similar glass transitions occur in higher dimensions when the right cost function is chosen. We also present some results for the statistics of distances for Poisson spatial point processes.