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

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 $R[\ln (N)\mathrm{/}c{]}^{1\mathrm{/}u}$, R is the root-mean-square displacement of a single random walker, and u and c determine how fast $1-{\mathrm{\Gamma }}_t(\mathbf{r})$ (the probability that site $\mathbf{r}$ has been visited by a single random walker by time t) decays for large values of r/R: $1-\Gamma_t({\bf r})\sim \exp[-c(r/R)^u]$\mathrm{.}Forthedeterministicfractalsconsideredinthispaper,$ u =d_w/(d_w-1)$,$d_w$beingtherandomwalkdimension\mathrm{.}Thecorrectiveterm$\Delta$isexpressedasaseriesin$\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 $\mathrm{\Delta }$ contributes substantially to the final value of $S_N(t)$ even for relatively large values of N.