### Abstract

In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If L_{n} be the number of the inner boundary points of random walk range in the n steps, we prove lim n→∞(L _{n}/n) exists with probability one. Also, we obtain some large deviation result for transient walk. We find that the expectation of the number of the inner boundary points of simple random walk on the two dimensional square lattice is of the same order as n/(log n)^{2}.

Original language | English |
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Pages (from-to) | 939-959 |

Number of pages | 21 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 68 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**The inner boundary of random walk range.** / Okada, Izumi.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 68, no. 3, pp. 939-959. https://doi.org/10.2969/jmsj/06830939

}

TY - JOUR

T1 - The inner boundary of random walk range

AU - Okada, Izumi

PY - 2016/1/1

Y1 - 2016/1/1

N2 - In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If Ln be the number of the inner boundary points of random walk range in the n steps, we prove lim n→∞(L n/n) exists with probability one. Also, we obtain some large deviation result for transient walk. We find that the expectation of the number of the inner boundary points of simple random walk on the two dimensional square lattice is of the same order as n/(log n)2.

AB - In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If Ln be the number of the inner boundary points of random walk range in the n steps, we prove lim n→∞(L n/n) exists with probability one. Also, we obtain some large deviation result for transient walk. We find that the expectation of the number of the inner boundary points of simple random walk on the two dimensional square lattice is of the same order as n/(log n)2.

UR - http://www.scopus.com/inward/record.url?scp=84979663342&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84979663342&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06830939

DO - 10.2969/jmsj/06830939

M3 - Article

AN - SCOPUS:84979663342

VL - 68

SP - 939

EP - 959

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -