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 upper bound on the exponent of covering probability of any such path when $d\ge 4$. This upper bound is asymptotically sharp for all monotonic paths. 查看全文>>