Late points for random walks in two dimensions

Let ${\mathcal{T}}_n(x)$ denote the time of first visit of a point $x$ on the lattice torus ${\mathbb{Z}}_n^2={\mathbb{Z}}^2\mathrm{/}n{\mathbb{Z}}^2$ by the simple random walk. The size of the set of $\alpha$, $n$-late points $$\mathcal{L}$_n(\alpha )={x\in \mathbb {Z}_n^2:$\mathcal{T}$_n(x)\geq \alpha \frac{4}{\pi}(n\log n)^2}$isapproximately$n^{2(1-\alpha)}$,for$\alpha \in (0,1)$[$$\mathcal{L}$_n(\alpha)$isemptyif$\alpha >1$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 $n^{2\beta (1-\alpha \mathrm{/}{\beta }^2)}$ 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.