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Probability Theory and Related Fields

, Volume 168, Issue 3–4, pp 691–719 | Cite as

Boundary of the range of transient random walk

  • Amine Asselah
  • Bruno SchapiraEmail author
Article
  • 199 Downloads

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.

Mathematics Subject Classification

60F05 60G50 

Notes

Acknowledgments

We would like to thank Gregory Maillard for discussions at an early stage of this work. We thank Pierre Mathieu for mentioning that we omitted to show that the limiting term in (1.14) was nonzero, and Perla Sousi for mentioning a few other inaccuracies. Finally, we thank an anonymous referee for his very careful reading, and his numerous corrections and suggestions which greatly improved the presentation. A.A. received support of the A\(^*\)MIDEX Grant (ANR-11-IDEX-0001-02) funded by the French Government “Investissements d’Avenir” program.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.LAMA, UPEC & IMéRACréteilFrance
  2. 2.Aix-Marseille Université, CNRS, Centrale MarseilleMarseilleFrance

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