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Boundary of the Range of Transient Random Walk (1507.01031v3)
Published 3 Jul 2015 in math.PR
Abstract: We study the boundary of the range of simple random walk on $\mathbb{Z}d$ in the transient regime $d\ge 3$. We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain a bound on the variance of order $n\log n$ in dimension three. We also establish a central limit theorem in dimension four and larger.