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January 2006 Late points for random walks in two dimensions
Amir Dembo, Yuval Peres, Jay Rosen, Ofer Zeitouni
Ann. Probab. 34(1): 219-263 (January 2006). DOI: 10.1214/009117905000000387

Abstract

Let $\mathscr{T}_{n}(x)$ 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 $ℒ_{n}(\alpha )=\{x\in \mathbb {Z}_{n}^{2}: \mathscr{T}_{n}(x)\geq \alpha \frac{4}{\pi}(n\log n)^{2}\}$ 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 $\beta <\sqrt{ \alpha }$), 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.

Citation

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Amir Dembo. Yuval Peres. Jay Rosen. Ofer Zeitouni. "Late points for random walks in two dimensions." Ann. Probab. 34 (1) 219 - 263, January 2006. https://doi.org/10.1214/009117905000000387

Information

Published: January 2006
First available in Project Euclid: 17 February 2006

zbMATH: 1100.60057
MathSciNet: MR2206347
Digital Object Identifier: 10.1214/009117905000000387

Subjects:
Primary: 60G50, 82C41
Secondary: 28A80

Rights: Copyright © 2006 Institute of Mathematical Statistics

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Vol.34 • No. 1 • January 2006
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