Scaling limits of tree-valued branching random walks

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}$! \geq ! 2$). 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, $\gamma \in (1,2]$. 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 $1\mathrm{/}2$ to move closer to the root of $\mathbb W_{ $\mathtt{b}$}$andprobability$1/(2$\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}$}$)withitsgraph-distance$d_{$\mathtt{g}\mathtt{r}$}$, then there exists a scaling sequence $(a_n{)}_{n\in \mathbb{N}}$ satisfying $a_n ! \rightarrow ! \infty$suchthatthemetricspace$(\mathcal R_{$\mathtt{b}$} (n), a_n^{-1}d_{$\mathtt{g}\mathtt{r}$})$, 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.