Frequently visited sites of the inner boundary of simple random walk range

This paper considers the question: how many times does a simple random walk revisit the most frequently visited site among the inner boundary points? It is known that in ℤ², the number of visits to the most frequently visited site among all of the points of the random walk range up to time n is asymptotic to π^-1(logn)², while in ℤ^d(d≥3), it is of order log n. We prove that the corresponding number for the inner boundary is asymptotic to β_dlogn for any d≥2, where β_d is a certain constant having a simple probabilistic expression.