Order statistics of Rosenstock's trapping problem in disordered media

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.