Scaling limits of critical branching random walks and their snakes

Abstract

We study scaling limits of $\mathbb R^d$-valued branching random walks (BRWs for short) in the following setting: indexing-trees are Galton-Watson trees with critical offspring distribution which are assumed, when appropriately rescaled, to converge to a critical $ψ$-Lévy tree; conditional on the indexing tree, the jumps of the BRW may exhibit dependence within the same sibling group, but distinct sibling groups are independent, and their distributions are centered but may vary (and for instance depend on the whole indexing tree), with a typical value of the walk nevertheless remaining within the domain of attraction of a mixture of Gaussian distributions. Under these assumptions and a necessary additional assumption, which we call the discrete Sheu assumption, we show that rescaled discrete snakes converge functionally to the $ψ$-Brownian snake introduced by Le Gall & Le Jan (1998) and D.~& Le Gall (2002). The method uses a coupling result for BRWs of independent interest. An application is given concerning the scaling limits of the range of BRWs that take their values in the $\mathtt b$-ary tree to the reflected $ψ$-Brownian cactus which is a variant of the Brownian cactus introduced by Curien, Le Gall & Miermont (2013).