Late points for random walks in two dimensions

Abstract

Let $\mathcal{T}_n(x)$ denote the time of first visit of a point $x$ on the lattice torus $\mathbb {Z}_n^2=\mathbb{Z}^2/n\mathbb{Z}^2$ by the simple random walk. The size of the set of $α$, $n$-late points $\mathcal{L}_n(α)=\{x\in \mathbb {Z}_n^2:\mathcal{T}_n(x)\geq α\frac{4}π(n\log n)^2\}$ is approximately $n^{2(1-α)}$, for $α\in (0,1)$ [$\mathcal{L}_n(α)$ is empty if $α>1$ and $n$ is large enough]. These sets have interesting clustering and fractal properties: we show that for $β\in (0,1)$, a disc of radius $n^β$ centered at nonrandom $x$ typically contains about $n^{2β(1-α/β^2)}$ points from $\mathcal{L}_n(α)$ (and is empty if $β<\sqrtα $), whereas choosing the center $x$ of the disc uniformly in $\mathcal{L}_n(α)$ boosts the typical number of $α, n$-late points in it to $n^{2β(1-α)}$. We also estimate the typical number of pairs of $α$, $n$-late points within distance $n^β$ of each other; this typical number can be significantly smaller than the expected number of such pairs, calculated by Brummelhuis and Hilhorst [Phys. A 176 (1991) 387--408]. On the other hand, our results show that the number of ordered pairs of late points within distance $n^β$ of each other is larger than what one might predict by multiplying the total number of late points, by the number of late points in a disc of radius $n^β$ centered at a typical late point.

Published at http://dx.doi.org/10.1214/009117905000000387 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)