## Annals of Probability

- Ann. Probab.
- Volume 34, Number 1 (2006), 219-263.

### Late points for random walks in two dimensions

Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni

#### Abstract

Let $\mathscr{T}_{n}(x)$ denote the time of first visit of a point *x* on the lattice torus ℤ_{n}^{2}=ℤ^{2}/*n*ℤ^{2} 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.

#### Article information

**Source**

Ann. Probab., Volume 34, Number 1 (2006), 219-263.

**Dates**

First available in Project Euclid: 17 February 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1140191537

**Digital Object Identifier**

doi:10.1214/009117905000000387

**Mathematical Reviews number (MathSciNet)**

MR2206347

**Zentralblatt MATH identifier**

1100.60057

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Secondary: 28A80: Fractals [See also 37Fxx]

**Keywords**

Planar random walk cover time late points multifractal analysis

#### Citation

Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer. Late points for random walks in two dimensions. Ann. Probab. 34 (2006), no. 1, 219--263. doi:10.1214/009117905000000387. https://projecteuclid.org/euclid.aop/1140191537