Key Takeaway: This paper presents a new calculation method that bridges the gap between short and long time frames, using both low- and high-density expansions in a novel two-point Pade approximant technique.

Abstract

connected nwlusters ' [u1 —W,i] connected decompositions [u1 —W,s] The problem of transport of a localized particle in a random network has been the subject of many investigations, ' ' which have been applicable either to very long or very short times. In this paper, we present a new calculation which bridges the gap between short and long times, by using both lowand high-density expansions in a novel two-point Pade approximant technique. " As an example, we compute the probability of remaining on the initial site, Po(r), as a function of time for a two-dimensional triangular lattice. Using scaling arguments based on the recent results of Coniglio for the structure of the percolating cluster, we examine both the critical and noncritical parts of Po(r) near the percolation edge, and finally we address the question of anomalous diffusion on this cluster. ' The model we consider is a triangular lattice with randomly populated sites, on which the transport is governed by a master equation with transfer matrix 8', whose elements W&= tv(g&. Here, w is the nonrandom nearest-neighbor jump rate and g, is a random variable for occupation of site i: (& = 0 if i is unoccupied, (;= 1 if it is occupied, and (g, ) = p (the concentration of occupied sites), where the bracket represents an average over all configurations consistent with p. The probability of occupation of a site at time t is governed by the averaged Green's function, G(t) = (exp Wr), or by its Laplace transform, G(u) = ([ul —W] '). The probability that the particle is at the initial site is given by Gas(t), which is, of course, a function of p. We write the low-density expansion of G (u ) as The first term on the right-hand side of Eq. (2) corresponds to solving the master equation on a given cluster of n sites; the second term removes from this all processes involving fewer than n sites, leaving only processes contributing to p". All possible locations of the n cluster are summed over in Eq. (2). To compute Gpp( ),uwe only consider connected clusters containing the origin and sum over all locations by taking the trace for each unique cluster. We enumerate the clusters and perform the matrix inversions on a computer. In the same spirit, we write the high-density expansion as