Random Walks on Lattices with Traps

Abstract

We consider random walks on simple cubic lattices containing two kinds of sites: ordinary ones and ``traps'' which, when stepped on, absorb the walker. We study two related problems: (a) the probability of returning to the origin and (b) the situation in which the particle can meet its end, not only by absorption at a trap, but also by a process, called spontaneous emission, which has a constant probability per step. In problem (b), we ask for the probability that emission, rather than absorption, occurs. The solution to (a) is known for 1 dimension, and given here for the 3-, 4-, … dimensional cases; the 2-dimensional case remains unsolved. The solution to (b) is known for the 1-, 3-, 4-, … dimensional cases; we give it for 2-dimensional case.