The Annals of Probability

Late points for random walks in two dimensions

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

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Let $\mathcal{T}_{n}(x)$ denote the time of first visit of a point x on the lattice torus ℤn2=ℤ2/n2 by the simple random walk. The size of the set of α, n-late points $\mathcal{L}_{n}(\alpha )=\{x\in \mathbb {Z}_{n}^{2}\dvtx \mathcal{T}_{n}(x)\geq \alpha \frac{4}{\pi}(n\log n)^{2}\}$ is approximately n2(1−α), for α∈(0,1) [ℒn(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 nβ centered at nonrandom x typically contains about n2β(1−α/β2) points from ℒn(α) (and is empty if $\beta \mbox{ \textless{} }\sqrt{ \alpha }$), whereas choosing the center x of the disc uniformly in ℒn(α) boosts the typical number of α,n-late points in it to n2β(1−α). We also estimate the typical number of pairs of α, n-late points within distance nβ 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 nβ 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 nβ centered at a typical late point.

Article information

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

First available in Project Euclid: 17 February 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Planar random walk cover time late points multifractal analysis


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.

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