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

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 (*) where N^D is the number of lattice points in D 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 with n in all numbers of dimensions. In D = 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid for N²≫N ≫ 1 when D = 1 and N ln N ≫n≫ 1 when D = 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.