Scaling limits of tree-valued branching random walks

We consider a branching random walk (BRW) taking its values in the b-ary rooted tree W-b (i.e. the set of finite words written in the alphabet {1, ..., b}, with b >= 2). The BRW is indexed by a critical Galton-Watson tree conditioned to have n vertices; its offspring distribution is aperiodic and is in the domain of attraction of a gamma-stable law, gamma is an element of (1, 2]. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on W-b (reflection at the root of W-b and otherwise: probability 1/2 to move closer to the root of W-b and probability 1/(2b) to move away from it to one of the b sites above). We denote by R-b (n) the range of the BRW in W-b which is the set of all sites in W-b visited by the BRW. We first prove a law of large numbers for #R-b (n) and we also prove that if we equip R-b (n) (which is a random subtree of W-b) with its graph-distance d(gr), then there exists a scaling sequence (a(n))(n is an element of N) satisfying a(n) -> infinity such that the metric space (R-b (n), a(n)(-1) d(gr)), equipped with its normalised empirical measure, converges to the reflected Brownian cactus with gamma-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by N. Curien, J-F. Le Gall and G. Miermont in [7].

Automated summary

This article discusses a branching random walk (BRW) on a b-ary rooted tree and proves various theoretical results on its properties. It also establishes a convergence relationship with a variant of the Brownian cactus.