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 $\mathcal{L}_n(\alpha )=\{x\in \mathbb {Z}_n^2:\mathcal{T}_n(x)\geq \alpha \frac{4}{\pi}(n\log n)^2\}$ 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.Comment: 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

Two-Dimensional Random Interlacements and Late Points for Random Walks

“…On the other hand, much remains to be discovered in two dimensions. After the heuristic arguments of [3] revealing an intriguing random set, it was shown in [8] that this set has interesting fractal-like properties when the elapsed time is a fraction of the expected cover time. This particular behaviour is induced by long distance correlations between hitting times due to recurrence.…”

Geometric structures of late points of a two-dimensional simple random walk

“…The cover time is the time taken to randomly walk in Z 2 n (= Z 2 /nZ 2 ) and visit every point of Z 2 n , and a late point of a random walk in Z 2 n is a point of Z 2 n , where the first hitting time is nearly equal to the cover time in a certain specific sense. We denote the set of α-late points in Z 2 n as L n (α) for 0 < α < 1 as in [10] (see (2.1) in the next section) and obtain certain asymptotic forms of |{ x ∈ L n (α) j : d(x i , x l ) ≤ n β for any 1 ≤ i, l ≤ j}| (1.1) for any 0 < α, β < 1 and j ∈ N, where x := (x 1 , . .…”

“…In addition, for 0 < α < 1, they used the set of α-high points in the GFF in Z 2 n (sites where the GFF takes high values) and α-favorite points in Z 2 (sites where the local time is close to that of the most frequently visited site). Dembo et al [10] and Brummelhuis and Hilhorst [3] estimated the number of pairs of α-late points, and Daviaud [4] estimated the α-high points. We show the corresponding results for the α-favorite points in our forthcoming paper.…”

Uniformity of the late points of random walk on $${\mathbb {Z}}_{n}^{d}$$ Z n d for $$d \ge 3$$ d ≥ 3

“…Our first main result is the following. In analogy with [9], we refer to the points in U(αt * ) as "α-late" for X . The reason for the terminology "late" is that the amount of time required by X to hit them is much larger than the maximal hitting time.…”

“…Thus the process of coverage in the two regimes is very different. The work [9] is part of a larger series which also includes [6][7][8] and the proofs of Theorems 1.1 and 1.2 employ several techniques which are present in the articles of this series.…”