Boundary of the Range of Transient Random Walk

Amine Asselah; Bruno Schapira

Boundary of the Range of Transient Random Walk

Probability 2016-06-10 v3

Authors: Amine Asselah , Bruno Schapira

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.

Keywords

transient random walk range boundary martingale central limit theorem variance bound

Cite

@article{arxiv.1507.01031,
  title  = {Boundary of the Range of Transient Random Walk},
  author = {Amine Asselah and Bruno Schapira},
  journal= {arXiv preprint arXiv:1507.01031},
  year   = {2016}
}

Comments

23 pages. Revised Version

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