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

Abstract

Let $b$ be an integer greater than 1 and let $W^{\ee}=(W^{\ee}_n; n\geq 0)$ be a random walk on the $b$-ary rooted tree $\U_b$, starting at the root, going up (resp. down) with probability $1/2+\epsilon$ (resp. $1/2 -\epsilon$), $\epsilon \in (0, 1/2)$, and choosing direction $i\in \{1, ..., b\}$ when going up with probability $a_i$. Here $\aa =(a_1, ..., a_b)$ stands for some non-degenerated fixed set of weights. We consider the range $\{W^{\ee}_n ; n\geq 0 \}$ that is a subtree of $\U_b $. It corresponds to a unique random rooted ordered tree that we denote by $\tau_{\epsilon}$. We rescale the edges of $\tau_{\epsilon}$ by a factor $\ee $ and we let $\ee$ 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 $\gamma (\aa)$. More precisely, we prove that $\tau_{\epsilon}$ converges to a continuum random tree encoded by two independent Brownian motions with drift conditioned to stay positive and scaled in time by $\gamma (\aa)$. 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 ($b=\infty$) and for a general set of weights $\aa =(a_n, n\geq 0)$.