Journal of the Mathematical Society of Japan

The inner boundary of random walk range


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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

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

First available in Project Euclid: 19 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

random walk range inner boundary ergodic theorem large deviation


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

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