2000/03/28 by L. Acedo, Acedo, L., S. B. Yuste +1
Physics and Astronomy · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.cond-mat/0003445
10 total pages (RevTex), 7 figures included
arxiv created 2000/03/28 · arxiv updated 2009/11/30
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 [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-Γt(\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-Γt(\bf r)∼ exp[-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 ln-n(N) lnm ln (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.