Journal of the Mathematical Society of Japan

The inner boundary of random walk range

Izumi OKADA

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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 $\lim_{n\to \infty} ({L_n}/{n})$ 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 ${n}/{(\log n)^2}$.

Article information

Source
J. Math. Soc. Japan Volume 68, Number 3 (2016), 939-959.

Dates
First available in Project Euclid: 19 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1468956153

Digital Object Identifier
doi:10.2969/jmsj/06830939

Mathematical Reviews number (MathSciNet)
MR3523532

Zentralblatt MATH identifier
06642398

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60F10: Large deviations

Keywords
random walk range inner boundary ergodic theorem large deviation

Citation

OKADA, Izumi. The inner boundary of random walk range. J. Math. Soc. Japan 68 (2016), no. 3, 939--959. doi:10.2969/jmsj/06830939. https://projecteuclid.org/euclid.jmsj/1468956153.


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