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 $x={[-\pi \ln \mathrm{}(1-c\left)\right]}^{\frac{2}{3}}{n}^{\frac{1}{3}}$ of the variables $n$ and $c$ only and is in leading order equal to $\left(\frac{8}{\pi }\right){\left(\frac{2}{3}\pi \right)}^{\frac{1}{2}}{x}^{\frac{3}{2}}\exp (-\frac{3x}{2})$. The analytical result includes correction terms and is confirmed by Monte Carlo simulations.