On the range of a constrained random walk

The authors study statistical properties of the range (¸= number of distinct sites visited) of a lattice random walk in discrete time constrained to visit a given site at a given time. In particular, they calculate the mean and obtain a bound on the variance of the range in the large time limit. The results are applied to a problem involving an unconstrained random walk in the presence of randomly distributed traps. A key role is played by the associated random walk that is obtained from the original random walk via a Cramér transform.