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 ${\mathbb{Z}}^2$, 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 $\pi^{-1}(\log n)^2$, while in ${\mathbb{Z}}^d$ $(d\ge3)$, it is of order $\log n$. We prove that the corresponding number for the inner boundary is asymptotic to $\beta_d\log n$ for any $d\ge2$, where $\beta_d$ is a certain constant having a simple probabilistic expression.