Scaling limits of tree-valued branching random walks

Abstract

We consider a branching random walk (BRW) taking its values in the b-ary rooted tree Wb (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 γ-stable law, γ∈(1,2]. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on Wb (reflection at the root of Wb and otherwise: probability 1∕2 to move closer to the root of Wb and probability 1∕(2b) to move away from it to one of the b sites above). We denote by Rb(n) the range of the BRW in Wb which is the set of all sites in Wb visited by the BRW. We first prove a law of large numbers for #Rb(n) and we also prove that if we equip Rb(n) (which is a random subtree of Wb) with its graph-distance dgr, then there exists a scaling sequence (an)n∈N satisfying an→∞ such that the metric space (Rb(n),an−1dgr), equipped with its normalised empirical measure, converges to the reflected Brownian cactus with γ-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].