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September 2015 The range of tree-indexed random walk in low dimensions
Jean-François Le Gall, Shen Lin
Ann. Probab. 43(5): 2701-2728 (September 2015). DOI: 10.1214/14-AOP947


We study the range $R_{n}$ of a random walk on the $d$-dimensional lattice $\mathbb{Z}^{d}$ indexed by a random tree with $n$ vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension $d\leq3$ that $n^{-d/4}R_{n}$ converges in distribution to the Lebesgue measure of the support of the integrated super-Brownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the tree-indexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching random walk in $\mathbb{Z}^{d}$, $d\leq3$. As an intermediate estimate, we get exact asymptotics for the probability that a critical branching random walk starting with a single particle at the origin hits a distant point. The results of the present article complement those derived in higher dimensions in our earlier work.


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Jean-François Le Gall. Shen Lin. "The range of tree-indexed random walk in low dimensions." Ann. Probab. 43 (5) 2701 - 2728, September 2015.


Received: 1 January 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1335.60070
MathSciNet: MR3395472
Digital Object Identifier: 10.1214/14-AOP947

Primary: 60G50 , 60J80
Secondary: 60G57

Keywords: Branching random walk , hitting probability , ISE , ‎range‎ , Super-Brownian motion , tree-indexed random walk

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 5 • September 2015
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