Order statistics of Rosenstock's trapping problem in disordered media

TL;DR

This paper studies the order statistics of trapping times for multiple random walkers in disordered media, revealing that higher-order Rosenstock approximations are necessary and effective for accurate modeling in such complex systems.

Contribution

It introduces a new approach to higher-order Rosenstock approximations based on the ratio of cumulants to moments, validated through simulations on disordered percolation aggregates.

Findings

Large ratios of cumulants to moments in disordered media

Higher-order Rosenstock approximations improve trapping time predictions

Simulation results confirm theoretical predictions

Abstract

The distribution of times $t_{j,N}$ elapsed until the first $j$ independent random walkers from a set of $N \gg 1$, all starting from the same site, are trapped by a quenched configuration of traps randomly placed on a disordered lattice is investigated. In doing so, the cumulants of the distribution of the territory explored by $N$ independent random walkers $S_N(t)$ and the probability $\Phi_N(t)$ that no particle of an initial set of $N$ is trapped by time $t$ are considered. Simulation results for the two-dimensional incipient percolation aggregate show that the ratio between the $n$th cumulant and the $n$th moment of $S_N(t)$ is, for large $N$, (i) very large in comparison with the same ratio in Euclidean media, and (ii) almost constant. The first property implies that, in contrast with Euclidean media, approximations of order higher than the standard zeroth-order Rosenstock approximation are required to provide a reasonable description of the trapping order statistics. Fortunately, the second property (which has a geometric origin) can be exploited to build these higher-order Rosenstock approximations. Simulation results for the two-dimensional incipient percolation aggregate confirm the predictions of our approach.