Late points for random walks in two dimensions

Let T-n(x) denote the time of first visit of a point x on the lattice torus Z(n)(2) = Z(2)/nZ(2) by the simple random walk. The size of the set of alpha, n-late points L-n(alpha) = {x is an element of Z(n)(2): T-n(x) >= alpha(4)/(pi)(n log n)(2)} is approximately n(2(1-alpha)), for alpha is an element of (0, 1) [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 is an element of (0, 1), a disc of radius n(beta) centered at nonrandom x typically contains about (n)2 beta(1-alpha/beta(2)) points from L-n(alpha) (and is empty if beta < root alpha), whereas choosing the center x of the disc uniformly in 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.