Random walk on fractals : numerical studies in two dimensions

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, we study: ( i ) the range RN of the walker (number of distinct visited sites during N steps): average value S, variance and asymptotic distribution; (ii) renewal theory (return to the original site): probability of return P,(N), mean number of returns v , ~ . 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/Orbach and Rammal/Toulouse).