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Thick points for intersections of planar sample paths


Authors: Amir Dembo, Yuval Peres, Jay Rosen and Ofer Zeitouni
Journal: Trans. Amer. Math. Soc. 354 (2002), 4969-5003
MSC (2000): Primary 60J55; Secondary 60J65, 28A80, 60G50
Published electronically: August 1, 2002
MathSciNet review: 1926845
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Abstract: Let $L_n^{X}(x)$ denote the number of visits to $x \in \mathbf{Z} ^2$ of the simple planar random walk $X$, up to step $n$. Let $X'$ be another simple planar random walk independent of $X$. We show that for any $0<b<1/(2 \pi)$, there are $n^{1-2\pi b+o(1)}$ points $x \in \mathbf{Z}^2$ for which $L_n^{X}(x)L_n^{X'}(x)\geq b^2 (\log n)^4$. This is the discrete counterpart of our main result, that for any $a<1$, the Hausdorff dimension of the set of thick intersection points $x$ for which $\limsup_{r \rightarrow 0} \mathcal{I} (x,r)/(r^2\vert\log r\vert^4)=a^2$, is almost surely $2-2a$. Here $\mathcal{I}(x,r)$ is the projected intersection local time measure of the disc of radius $r$ centered at $x$ for two independent planar Brownian motions run until time $1$. The proofs rely on a ``multi-scale refinement'' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius $r$centered at $x$ by $x+rK$ for general sets $K$.


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Additional Information

Amir Dembo
Affiliation: Departments of Mathematics and Statistics, Stanford University, Stanford, California 94305
Email: amir@math.stanford.edu

Yuval Peres
Affiliation: Department of Statistics, University of California Berkeley, Berkeley, California 94720 and Institute of Mathematics, Hebrew University, Jerusalem, Israel
Email: peres@stat.berkeley.edu

Jay Rosen
Affiliation: Department of Mathematics, College of Staten Island, CUNY, Staten Island, New York 10314
Email: jrosen3@earthlink.net

Ofer Zeitouni
Affiliation: Department of Electrical Engineering, Technion, Haifa 32000, Israel
Email: zeitouni@ee.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9947-02-03080-5
Keywords: Thick points, intersection local time, multi-fractal analysis, stable process
Received by editor(s): May 9, 2001
Received by editor(s) in revised form: April 16, 2002
Published electronically: August 1, 2002
Additional Notes: The first author’s research was partially supported by NSF grant #DMS-0072331
The second author’s research was partially supported by NSF grant #DMS-9803597
The third author’s research was supported, in part, by grants from the NSF and from PSC-CUNY
The research of all authors was supported, in part, by a US-Israel BSF grant
Article copyright: © Copyright 2002 American Mathematical Society