Exploratory behavior, trap models, and glass transitions

Abstract

A random walk is performed on a disordered landscape composed of N sites randomly and uniformly distributed inside a d-dimensional hypercube. The walker hops from one site to another with probability proportional to exp[−βE(D)], where β=1/T is the inverse of a formal temperature and E(D) is an arbitrary cost function which depends on the hop distance D. Analytic results indicate that, if E(D)=Dd and N→∞, there exists a glass transition at βd=nd/2/[(d/2)r(d/2)]. Below Td, the average trapping time diverges and the system falls into an out-of-equilibrium regime with aging phenomena. A Lévy flight scenario and applications of exploratory behavior are considered.