On covering monotonic paths with simple random walk

In this paper we study the probability that a d dimensional simple random walk (or the first L steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an L₁ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when d ≥ 4. The main tool is a general combinatorial inequality, that is interesting in its own right.