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

Abstract

Let b be an integer greater than 1 and let W^ɛ=(W^ɛ_n;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 a_i. Here a=(a₁,…,a_b) stands for some nondegenerated fixed set of weights. We consider the range {W^ɛ_n;n≥0} that is a subtree of $\mathbb {U}_{b}$. It corresponds to a unique random rooted ordered tree that we denote by τ_ɛ. We rescale the edges of τ_ɛ 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 τ_ɛ 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).