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 Z2Z2, the number of visits to the most frequently visited site among all of the points of the random walk range up to time nn is asymptotic to π−1(logn)2π−1(logn)2, while in ZdZd(d≥3)(d≥3), it is of order lognlogn. We prove that the corresponding number for the inner boundary is asymptotic to βdlognβdlogn for any d≥2d≥2, where βdβd is a certain constant having a simple probabilistic expression.