Late points for random walks in two dimensions

Abstract

Let $\mathscr{T}_{n}(x)$ denote the time of first visit of a point x on the lattice torus ℤ_n²=ℤ²/nℤ² by the simple random walk. The size of the set of α, n-late points $ℒ_{n}(\alpha )=\{x\in \mathbb {Z}_{n}^{2}: \mathscr{T}_{n}(x)\geq \alpha \frac{4}{\pi}(n\log n)^{2}\}$ is approximately n^2(1−α), for α∈(0,1) [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∈(0,1), a disc of radius n^β centered at nonrandom x typically contains about n^{2β(1−α/β2)} points from ℒ_n(α) (and is empty if $\beta <\sqrt{ \alpha }$), whereas choosing the center x of the disc uniformly in 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.