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

Abstract

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)∼( ln t) γ(ε) where γ( ε) depends on the strength ε of the field. The probability P( r, t) to find the random walker after t time-steps on a site at distance r from its starting point decays asymptotically as − ln P(r,t)∼−r u(t) with u( t)=ln( t)/(ln( t)− γ( ε)) and approaches a simple exponential for asymptotic large time. A similiar picture arises for the behavior of the probability density P(ℓ, t), where ℓ is the chemical (shortest path) distance from the origin of the random walk.