We consider a branching random walk (BRW) taking its values in the -ary rooted tree (i.e. the set of finite words written in the alphabet , with ). 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, . The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on (reflection at the root of and otherwise: probability to move closer to the root of and probability to move away from it to one of the sites above). We denote by the range of the BRW in which is the set of all sites in visited by the BRW. We first prove a law of large numbers for and we also prove that if we equip (which is a random subtree of ) with its graph-distance , then there exists a scaling sequence satisfying such that the metric space , 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 .
N. Torri was supported by the project Labex MME-DII (ANR11-LBX-0023-01).
The authors would like to thank the referee and the editor for their helpful suggestions.
"Scaling limits of tree-valued branching random walks." Electron. J. Probab. 27 1 - 54, 2022. https://doi.org/10.1214/22-EJP741