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

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We address the problem of evaluating the number Sₙ(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, Sₙ(t) ≈ Sₙ(t) (1-Δ), where Sₙ(t) is the number of sites inside a hypersphere of radius R [(N)/c]¹/ ᵘ, R is the root-mean-square displacement of a single random walker, and u and c determine how fast 1-Γₜ(r) (the probability that site r has been visited by a single random walker by time t) decays for large values of r/R: 1-Γₜ(r) [-c(r/R)ᵘ]. For the deterministic fractals considered in this paper, u =dw/(dw-1), dw being the random walk dimension. The corrective term Δ is expressed as a series in ⁻ⁿ(N) ᵐ (N) (with n≥ 1 and 0≤ m≤ 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 Sₙ(t) even for relatively large values of N.