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

摘要

We address the problem of evaluating the number $S_N(t)$ of distinct sitesvisited up to time t by N noninteracting random walkers all initially placed onone site of a deterministic fractal lattice. For a wide class of fractals, ofwhich the Sierpinski gasket is a typical example, we propose that, after theshort-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 hypersphereof radius $R [\ln (N)/c]^{1/ u}$, R is the root-mean-square displacement of asingle random walker, and u and c determine how fast $1-\Gamma_t({\bf r})$ (theprobability that site ${\bf r}$ has been visited by a single random walker bytime t) decays for large values of r/R: $1-\Gamma_t({\bf r})\sim\exp[-c(r/R)^u]$. For the deterministic fractals considered in this paper, $ u=d_w/(d_w-1)$, $d_w$ being the random walk dimension. The corrective term$\Delta$ is expressed as a series in $\ln^{-n}(N) \ln^m \ln (N)$ (with $n\geq1$ and $0\leq m\leq n$), which is given explicitly up to n=2. Numericalsimulations on the Sierpinski gasket show reasonable agreement with theanalytical expressions. The corrective term $\Delta$ contributes substantiallyto the final value of $S_N(t)$ even for relatively large values of N.