Random walk on fractals: numerical studies in two dimensions

Abstract

Monte Carlo calculations are used to investigate some statistical properties of random walks on fractal structures. Two kinds of lattices are used: the Sierpinski gasket and the infinite percolation cluster, in two dimensions. Among other problems, the authors study: (i) the range R_N of the walker (number of distinct visited sites during N steps): average value S_N, variance sigma _N and asymptotic distribution: (ii) renewal theory (return to the original site): probability of return P₀(N), mean number of returns nu _N. The probability distribution of the walker position P(N,R) after N steps is discussed. The asymptotic behaviour (N>>1) of these quantities exhibits power laws, with associated exponents. The numerical values of these exponents are in good agreement with recent theoretical predictions (Alexander and Orbach, 1982; Rammal and Toulouse, 1982).