The inner boundary of random walk range

2014/07/08 by Izumi Okada, Okada, Izumi
Decision Sciences · Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1407.2081

openalex publication_date 2014/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 two dimensionnal square lattice is of the same order as (n)/((log n)²).

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