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, $S_N(t)\approx {\hat{S}}_N(t)(1-\Delta )$, 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 $1-{\Gamma }_t(\mathbf{r})$ (the probability that site $\mathbf{r}$ has been visited by a single random walker by time t) decays for large values of r/R: $1-{\Gamma }_t(\mathbf{r})\sim \exp [-c(r/R{)}^u]$. For the deterministic fractals considered in this paper, $u=d_w/(d_w-1)$, $d_w$ being the random walk dimension. The corrective term $\Delta$ is expressed as a series in ${\ln }^{-n}(N){\ln }^m\ln (N)$ (with $n\ge 1$ and $0\le m\le n$), which is given explicitly up to n=2. Numerical simulations on the Sierpinski gasket show reasonable agreement with the analytical expressions. The corrective term $\Delta$ contributes substantially to the final value of $S_N(t)$ even for relatively large values of N.

10 total pages (RevTex), 7 figures included