Journal of Statistical Physics, VoL 34, Nos. 3/4, 1984 Diffusion on Random Systems above, below, and at Their Percolation Threshold in Two and Three Dimensions R. B. Pandey, 1'2 D. Stauffer, 1 A. Margolina, 3 and J. G. Zabolitzky I Received September 22, 1983 A detailed Monte Carlo study is presented for classical diffusion (random walks) on random L * L triangular and L * L * L simple cubic lattices, with L up to 4096 and 256, respectively. The speed of a Cyber 205 vector computer is found to be about one order of magnitude larger than that of a usual CDC Cyber 76 computer. To reach the asymptotic scaling regime, walks with up to 10 million steps were simulated, with about 10 jl steps in total for L = 256 at the percola- tion threshold. We review and extend the dynamical scaling description for the distance traveled as function of time, the diffusivity above the threshold, and the cluster radius below. Earlier discrepancies between scaling theory and computer experiment are shown to be due to insufficient Monte Carlo data. The conduc- tivity exponent /~ is found to be 2.0 _+0.2 in three and 1.28 +_ 0.02 in two dimensions. Our data in three dimensions follow well the finite-size scaling theory. Below the threshold, the approach of the distance traveled to its asymptotic value is consistent with theoretical speculations and an exponent 2/5 independent of dimensionality. The correction-to-scaling exponent at Pc seems to be larger in two than in three dimensions. KEY WORDS: Random walk; percolation; conductivity; scaling; fractals; Monte Carlo; FORTRAN programs; supercomputer. 1. INTRODUCTION The "ant in the labyrinth" of de Gennes (1) has become a conventional term to discuss the problem of classical diffusion in random percolating sys- tems. (2-8) Here one studies the random walk motion of particles on the 1 Institute of Theoretical Physics, Cologne University, 5000 K61n 41, West Germany. 2 Now at the University of Cambridge, Department of Physics, Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, U.K. 3 Now at Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544, USA. 427 0022-4715/84/0200-0427503.50/0 9 1984 Plenum Publishing Corporation