Random walks on percolation with a topological bias:: Decay of the probability density

We investigate random walks on the infinite percolation cluster at the critical concentration p(c) under the influence of a topological bias field, where the hopping rates towards larger chemical distances l from the origin of the walk are increased, We find that the root mean square displacement evolves with time as R(t) similar to (In t)(gamma(epsilon)) where (gamma(epsilon)) depends on the strength epsilon of the field. The probability P(r, t) to find the random walker after t time-steps on a sire at distance r from its starting point decays asymptotically as -ln P(r,t)similar to -r(u(t)) with u(t)= ln(t)/(ln(t) - gamma(epsilon)) and approaches a simple exponential for asymptotic large time. A similiar picture arises for the behavior of the probability density P(l, t), where e is the chemical (shortest path) distance from the origin of the random walk. (C) 1999 Elsevier Science B.V. All rights reserved.