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 $\exp [- \beta E(D)]$, 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 L\'evy flight scenario and applications to exploratory behavior are considered.