Escaping from cycles through a glass transition

TL;DR

This paper investigates how a random walk in a disordered medium undergoes a glass transition at a critical temperature, leading to ergodicity breaking and trapping in attractor basins, with analytical results in one dimension and extensions to higher dimensions.

Contribution

It introduces a model of random walks with distance-dependent transition costs, revealing a glass transition and ergodicity breaking in disordered media across dimensions.

Findings

Glass transition at T=1/2 in 1D as N→∞

Divergence of trapping times below T_1

Existence of similar transitions in higher dimensions

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 ${\cal O}(N)$ 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.