Scaling limits of tree-valued branching random walks

Abstract

We consider a branching random walk (BRW) taking its values in the $\mathtt{b}$-ary rooted tree ${\mathbb{W}}_{\mathtt{b}}$ (i.e. the set of finite words written in the alphabet $\{1,\dots ,\mathtt{b}\}$, with $\mathtt{b}\ge 2$). 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, $\mathit{\gamma }\in (1,2]$. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on ${\mathbb{W}}_{\mathtt{b}}$ (reflection at the root of ${\mathbb{W}}_{\mathtt{b}}$ and otherwise: probability $1∕2$ to move closer to the root of ${\mathbb{W}}_{\mathtt{b}}$ and probability $1∕(2\mathtt{b})$ to move away from it to one of the $\mathtt{b}$ sites above). We denote by ${\mathcal{R}}_{\mathtt{b}}(n)$ the range of the BRW in ${\mathbb{W}}_{\mathtt{b}}$ which is the set of all sites in ${\mathbb{W}}_{\mathtt{b}}$ visited by the BRW. We first prove a law of large numbers for $\mathrm{\#}{\mathcal{R}}_{\mathtt{b}}(n)$ and we also prove that if we equip ${\mathcal{R}}_{\mathtt{b}}(n)$ (which is a random subtree of ${\mathbb{W}}_{\mathtt{b}}$) with its graph-distance $d_{\mathtt{gr}}$, then there exists a scaling sequence ${(a_n)}_{n\in \mathbb{N}}$ satisfying $a_n\to \mathrm{\infty }$ such that the metric space $({\mathcal{R}}_{\mathtt{b}}(n),a_n^{-1}{d}_{\mathtt{gr}})$, 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].