Boundary of the range of transient random walk

Abstract

We study the boundary of the range of simple random walk on $$\mathbb {Z}^d$$ in the transient case $$d\ge 3$$ . We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain an upper bound on the variance of order $$n\log n$$ in dimension three. We also establish a Central Limit Theorem in dimension four and larger.