Asymmetric simple exclusion process on the percolation cluster: Waiting time distribution in side-branches

Abstract

As the simplest model of transport of interacting particles in a disordered medium, we consider the asymmetric simple exclusion process (ASEP) in which particles with hard-core interactions perform biased random walks, on the supercritical percolation cluster. In this process, the long time trajectory of a marked particle consists of steps on the backbone, punctuated by time spent in side-branches. We study the probability distribution in the steady state of the waiting time $T_w$ of a randomly chosen particle, in a side-branch since its last step along the backbone. Exact numerical evaluation of this on a single side-branch of length $L=$ $1$ to $9$ shows that for large fields, the probability distribution of $\log T_w $ has multiple well separated peaks. We extend this result to a regular comb, and to the ASEP on the percolation cluster. We show that in the steady state, the fractional number of particles that have been in the same side-branch for a time interval greater than $T_w$ varies as $exp( - c \sqrt{\log T_w})$ for large $T_w$, where $c$ depends only on the bias field. However, these long timescales are not reflected in the eigenvalue spectrum of the Markov evolution matrix. The system shows dynamical heterogeneity, with particles segregating into pockets of high and low mobilities.