Open Access
July, 2016 The inner boundary of random walk range
J. Math. Soc. Japan 68(3): 939-959 (July, 2016). DOI: 10.2969/jmsj/06830939


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 ${n}/{(\log n)^2}$.


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Izumi OKADA. "The inner boundary of random walk range." J. Math. Soc. Japan 68 (3) 939 - 959, July, 2016.


Published: July, 2016
First available in Project Euclid: 19 July 2016

zbMATH: 1359.60061
MathSciNet: MR3523532
Digital Object Identifier: 10.2969/jmsj/06830939

Primary: 60J05
Secondary: 60F10

Keywords: ergodic theorem , inner boundary , large deviation , random walk range

Rights: Copyright © 2016 Mathematical Society of Japan

Vol.68 • No. 3 • July, 2016
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