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 $\alpha$, $n$-late points ${\mathcal{L}}_n(\alpha )=\{x\in {\mathbb{Z}}_n^2:{\mathcal{T}}_n(x)\ge \alpha \frac{4}{\pi }(n\log n{)}^2\}$ is approximately $n^{2(1-\alpha )}$, for $\alpha \in (0,1)$ [${\mathcal{L}}_n(\alpha )$ is empty if $\alpha >1$ and $n$ is large enough]. These sets have interesting clustering and fractal properties: we show that for $\beta \in (0,1)$, a disc of radius $n^{\beta }$ centered at nonrandom $x$ typically contains about $n^{2\beta (1-\alpha /{\beta }^2)}$ points from ${\mathcal{L}}_n(\alpha )$ (and is empty if $\beta <\sqrt{\alpha }$), whereas choosing the center $x$ of the disc uniformly in ${\mathcal{L}}_n(\alpha )$ boosts the typical number of $\alpha ,n$-late points in it to $n^{2\beta (1-\alpha )}$. We also estimate the typical number of pairs of $\alpha$, $n$-late points within distance $n^{\beta }$ 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 $n^{\beta }$ 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 $n^{\beta }$ 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)

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