II. Territory covered by N random walkers on stochastic fractals. The percolation aggregate

Abstract

The average number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in a disordered fractal is considered. This quantity $S_N(t)$ is the result of a double average: an average over random walks on a given lattice followed by an average over different realizations of the lattice. We show for two-dimensional percolation clusters at criticality (and conjecture for other stochastic fractals) that the distribution of the survival probability over these realizations is very broad in Euclidean space but very narrow in the chemical or topological space. This allows us to adapt the formalism developed for Euclidean and deterministic fractal lattices to the chemical language, and an asymptotic series for $S_N(t)$ analogous to that found for the non-disordered media is proposed here. The main term is equal to the number of sites (volume) inside a ``hypersphere'' in the chemical space of radius $L [\ln (N)/c]^{1/v}$ where L is the root-mean-square chemical displacement of a single random walker, and v and c determine how fast $1-Γ_t(\ell)$ (the probability that a given site at chemical distance $\ell$ from the origin is visited by a single random walker by time t) decays for large values of $\ell/L$: $1-Γ_t(\ell)\sim \exp[-c(\ell/L)^v]$. The parameters appearing in the first two asymptotic terms of $S_N(t)$ are estimated by numerical simulation for the two-dimensional percolation cluster at criticality. The corresponding theoretical predictions are compared with simulation data, and the agreement is found to be very good.