The inner boundary of random walk range

In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If $L_n$ be the number of the inner boundary points of random walk range in the $n$ steps, we prove $\lim_{n\to \infty}\frac{L_n}{n}$ exists with probability one. Also, we obtain some large deviation result for transient walk. We find that the expectation of the number of the inner boundary points of simple random walk on two dimensionnal square lattice is of the same order as $\frac{n}{(\log n)^2}$.