Multiparticle trapping problem in the half-line

Abstract

A variation of Rosenstock's trapping model in which $N$ independent random walkers are all initially placed upon a site of a one-dimensional lattice in the presence of a {\em one-sided} random distribution (with probability $c$) of absorbing traps is investigated. The probability (survival probability) $\Phi_N(t)$ that no random walker is trapped by time $t$ for $N \gg 1$ is calculated by using the extended Rosenstock approximation. This requires the evaluation of the moments of the number $S_N(t)$ of distinct sites visited in a {\em given} direction up to time $t$ by $N$ independent random walkers. The Rosenstock approximation improves when $N$ increases, working well in the range $Dt\ln^2(1-c) \ll \ln N$, $D$ being the diffusion constant. The moments of the time (lifetime) before any trapping event occurs are calculated asymptotically, too. The agreement with numerical results is excellent.