ESPCI web site
Universal Exploration Dynamics of Random Walks
The territory explored by a random walk is a key property that may be quantified by the number of distinct sites that the random walk visits up to a given time. We recently introduced a more fundamental quantity, the time $\tau_n$ required by a random walk to find a site that it never visited previously when the walk has already visited $n$ distinct sites, which encompasses the full dynamics about the visitation statistics. To study it, we develop a theoretical approach that relies on a mapping with a trapping problem, in which the spatial distribution of traps is continuously updated by the random walk itself. Despite the geometrical complexity of the territory explored by a random walk, the distribution of the $\tau_n$ can be accounted for by simple analytical expressions. Processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes.