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November 2005 Continuum tree limit for the range of random walks on regular trees
Thomas Duquesne
Ann. Probab. 33(6): 2212-2254 (November 2005). DOI: 10.1214/009117905000000468

Abstract

Let b be an integer greater than 1 and let Wɛ=(Wɛn;n≥0) be a random walk on the b-ary rooted tree $\mathbb {U}_{b}$, starting at the root, going up (resp. down) with probability 1/2+ɛ (resp. 1/2−ɛ), ɛ∈(0,1/2), and choosing direction i∈{1,…,b} when going up with probability ai. Here a=(a1,…,ab) stands for some nondegenerated fixed set of weights. We consider the range {Wɛn;n≥0} that is a subtree of $\mathbb {U}_{b}$. It corresponds to a unique random rooted ordered tree that we denote by τɛ. We rescale the edges of τɛ by a factor ɛ and we let ɛ go to 0: we prove that correlations due to frequent backtracking of the random walk only give rise to a deterministic phenomenon taken into account by a positive factor γ(a). More precisely, we prove that τɛ converges to a continuum random tree encoded by two independent Brownian motions with drift conditioned to stay positive and scaled in time by γ(a). We actually state the result in the more general case of a random walk on a tree with an infinite number of branches at each node (b=∞) and for a general set of weights a=(an,n≥0).

Citation

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Thomas Duquesne. "Continuum tree limit for the range of random walks on regular trees." Ann. Probab. 33 (6) 2212 - 2254, November 2005. https://doi.org/10.1214/009117905000000468

Information

Published: November 2005
First available in Project Euclid: 7 December 2005

zbMATH: 1099.60021
MathSciNet: MR2184096
Digital Object Identifier: 10.1214/009117905000000468

Subjects:
Primary: 05C05, 05C80, 60F17, 60J80

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 6 • November 2005
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