I. Territory covered by N random walkers on deterministic fractals. The Sierpinski gasket

Abstract

We address the problem of evaluating the number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all initially placed on one site of a deterministic fractal lattice. For a wide class of fractals, of which the Sierpinski gasket is a typical example, we propose that, after the short-time compact regime and for large N, $S_N(t) \approx \hat{S}_N(t) (1-Δ)$, where $\hat{S}_N(t)$ is the number of sites inside a hypersphere of radius $R [\ln (N)/c]^{1/ u}$, R is the root-mean-square displacement of a single random walker, and u and c determine how fast $1-Γ_t({\bf r})$ (the probability that site ${\bf r}$ has been visited by a single random walker by time t) decays for large values of r/R: $1-Γ_t({\bf r})\sim \exp[-c(r/R)^u]$. For the deterministic fractals considered in this paper, $ u =d_w/(d_w-1)$, $d_w$ being the random walk dimension. The corrective term $Δ$ is expressed as a series in $\ln^{-n}(N) \ln^m \ln (N)$ (with $n\geq 1$ and $0\leq m\leq n$), which is given explicitly up to n=2. Numerical simulations on the Sierpinski gasket show reasonable agreement with the analytical expressions. The corrective term $Δ$ contributes substantially to the final value of $S_N(t)$ even for relatively large values of N.