Field-induced drift and trapping in percolation networks

Abstract

The authors study the effects of a bias-producing external field on a random walk on the infinite cluster in the percolation problem. There are two competing physical effects (drift and trapping) which result in a drift velocity nu which rises and then falls as the field increases. They study these effects on a one-dimensional lattice with random-length branches, and on the diluted Bethe lattice. They calculate nu in the first model with a maximum allowed length for each branch. In the limit that this cutoff length becomes infinite, they find that the velocity vanishes identically above a finite critical value of the field. For the Bethe lattice, they derive an upper bound on the critical field, which varies as (p-p_c)^½ as the percolation concentration p_c is approached. In the one-dimensional model, they also investigate the anomalous regime in which the velocity vanishes. They discuss the distribution of steady state times required to traverse N sites, and find that it can be described in terms of a stable distribution of index x with superimposed oscillations. The index x of the stable distribution is given by L/ xi where xi is a characteristic branch length and L is a bias-induced length which describes the exponential buildup of the steady state density of particles towards the end of a branch.