The Annals of Probability

Continuum tree limit for the range of random walks on regular trees

Thomas Duquesne

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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).

Article information

Ann. Probab., Volume 33, Number 6 (2005), 2212-2254.

First available in Project Euclid: 7 December 2005

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C05: Trees 05C80: Random graphs [See also 60B20]

Continuum random tree contour process exploration process height process limit theorem random walk range regular tree


Duquesne, Thomas. Continuum tree limit for the range of random walks on regular trees. Ann. Probab. 33 (2005), no. 6, 2212--2254. doi:10.1214/009117905000000468.

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