  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 $L_1$ 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 $d\ge 4$.
