Exploratory Behavior, Trap Models and Glass Transitions

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 $E(D) = D^{d}$ and $N \to \infty$, there exists a glass transition at $β_d = π^{d/2}/Γ(d/2 + 1)$. 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.

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