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

Abstract

Let b be an integer greater than 1 and let W ε = ( W nε;n ≥ 0) be a random walk on the b-ary rooted tree double-struck U sign b, 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 a i. Here a = (a 1,..., a b) stands for some nondegenerated fixed set of weights. We consider the range {W nε; n ≥ 0} that is a subtree of double-struck U sign b. It corresponds to a unique random rooted ordered tree that we denote by τ ε. We rescale the edges of tau; 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 = (a n, n > 0). © Institute of Mathematical Statistics, 2005.