On the survival probability of a random walk in a finite lattice with a single trap

Abstract

We consider the survival of a random walker in a finite lattice with periodic boundary conditions. The initial position of the random walker is uniformly distributed on the lattice with respect to the trap. We show that the survival of a random walker, 〈U _n>, can be exactly related to the expected number of distinct sites visted on a trap-free lattice by 〈U _n〉=1−〈S _n〉/N ^D (*) whereN ^D is the number of lattice points inD dimensions. We then analyze the behavior of 〈S_n〉 in any number of dimensions by using Tauberian methods. We find that at sufficiently long times 〈S _n〉 decays exponentially withn in all numbers of dimensions. InD = 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forN ²≫N ≫ 1 whenD = 1 andN lnN ≫n≫ 1 whenD = 2. No such crossover exists when Z⩾3. The form of (*) suggests that the single trap approximation is indeed a valid low-concentration limit for survival on an infinite lattice with a finite concentration of traps.