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 $\mathcal{O}(N)$ number of attractor basins of two-cycles that trap the walker. For $d=1$, analytic results indicate the existence of a glass transition at $T_1=1/2$ as $N\to \infty$. Below $T_1$, the average trapping time in two-cycles diverges and out-of-equilibrium behavior appears. Similar glass transitions occur in higher dimensions choosing a proper cost function. We also present some results for the statistics of distances for Poisson spatial point processes.

11 pages, 4 figures

References in corpus (5)

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