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

We investigate random walks on the infinite percolation cluster at the crit ical concentration p(c) under the influence of a topological bias field, wh ere the hopping rates towards larger chemical distances l from the origin o f the walk are increased, We find that the root mean square displacement ev olves with time as R(t) similar to (In t)(gamma(epsilon)) where (gamma(epsi lon)) 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)) w ith u(t)= ln(t)/(ln(t) - gamma(epsilon)) and approaches a simple exponentia l 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) di stance from the origin of the random walk. (C) 1999 Elsevier Science B.V. A ll rights reserved.