Asymptotically Exact Solution of the One-Dimensional Trapping Problem

The survival probability of a particle that performs a random walk on a linear chain with randomly distributed traps is considered. An asymptotically exact solution was found, valid for large step numbers n and all trap concentrations c. The survival probability depends upon the combination x=[-πln(1-c)]2/3n1/3 of the variables n and c only and is in leading order equal to (8π)(2/3π)1/2x3/2exp(-3x/2). The analytical result includes correction terms and is confirmed by Monte Carlo simulations.