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 $\exp [-\beta E(D)]$, where $\beta =1/T$ 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 $E(D)=D^d$ and $N\to \infty$, there exists a glass transition at ${\beta }_d={\pi }^{d/2}/\Gamma (d/2+1)$. Below $T_d$, the average trapping time diverges and the system falls into an out-of-equilibrium regime with aging phenomena. A Lévy flight scenario and applications to exploratory behavior are considered.

4 pages, 1 figure, new version

References in corpus (6)

• Sub-diffusion and localization in the one dimensional trap model
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• Deterministic walks in random networks: an application to thesaurus graphs
• Escaping from cycles through a glass transition
• Statistical Physics of the Glass Phase
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