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Continuum Tree Limit for the Range of Random Walks on Regular Trees

Thomas Duquesne
The Annals of Probability
Vol. 33, No. 6 (Nov., 2005), pp. 2212-2254
Stable URL: http://www.jstor.org/stable/3481782
Page Count: 43
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Continuum Tree Limit for the Range of Random Walks on Regular Trees
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Abstract

Let b be an integer greater than 1 and let Wε=(Wn ε;n≥ 0) be a random walk on the b-ary rooted tree Ub, starting at the root, going up (resp. down) with probability 1/2 + ε (resp. 1/2 - ε), ε ∈ (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 {Wn ε;n≥ 0} that is a subtree of Ub. It corresponds to a unique random rooted ordered tree that we denote by τ ε. We rescale the edges of τ ε by a factor ε and we let ε 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 τ ε 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).

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