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Boundary of the range of transient random walk. (English) Zbl 1382.60066
The authors of this paper study the boundary of the range of a simple random walk on \(\mathbb Z^d\) in the transient case \(d >2\). The authors show (Proposition 1.2) that volumes of the range and its boundary differ mainly by a martingale. Proposition 1.4 presents a precise estimate on the mean square of the martingale. This estimate is delicate, uses precise Green function asymptotics, and symmetry of the walk. As a consequences of Propositions 1.2 and 1.4, the authors obtain an upper bound on the variance of order \(n\log n\) in dimension three and establish a central limit theorem in dimension four and larger.

MSC:
60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
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