Counting planar random walk holes

Abstract

We study two variants of the notion of holes formed by planar simple random walk of time duration $2n$ and the areas associated with them. We prove in both cases that the number of holes of area greater than $A(n)$, where $\{A(n)\}$ is an increasing sequence, is, up to a logarithmic correction term, asymptotic to $n\cdot A(n)^{-1}$ for a range of large holes, thus confirming an observation by Mandelbrot. A consequence is that the largest hole has an area which is logarithmically asymptotic to $n$. We also discuss the different exponent of 5/3 observed by Mandelbrot for small holes.