Long-range random walk on percolation clusters

Random walks on square-lattice percolation clusters are simulated for interaction ranges spanning one to five nearest-neighbor bonds (R=1 , . . . , 5). The relative hopping probability is given by exp(-αr), where r is the number of bonds traversed in one hop and α is a parameter (0≤α≤10). The fractal exponent for the random walks is universal. For R=2 (and R=1) we obtain a spectral dimension of $d_s$=1.31±0.03, in agreement with the Alexander-Orbach conjecture (1.333), and in even better agreement with the Aharony-Stauffer conjecture (1.309). Our results are based on the relation $d_s$=(91/43)f, where $S_n$∼$N^f$ describes the mean number ($S_N$) of distinct sites visited in N steps for walks originating on all clusters. While the asymptotic limit of f is closely approached after 5000 nominal time steps for α=0, much longer times (>50 000 steps) are required for α≫0. We also observe fractal-to-Euclidean crossovers above criticality; again, this crossover takes much longer for α≫0.