Scaling limit of the range of tree-valued branching random walks in random environmen

Abstract

We study a branching random walk (BRW) taking its values in a random tree $\bT$ (seen as a family tree) with an infinite line of ancestors that is a variant of a supercritical Galton--Watson (GW) tree with offspring distribution $\nu$. The transition probabilities of the BRW are those of a critical biased random walk on $\bT$: namely, the probability to move from $x$ to one of its $k_x$ children is $1/(\mathtt{m}_\nu+k_x)$ and the probability to move from $x$ to the direct parent of $x$ is $\mathtt{m}_\nu/(\mathtt{m}_\nu+k_x)$. Here $\ttm_\nu$ stands for the mean of $\nu$. The BRW is indexed by a critical GW tree conditioned to have $n$ {vertices} and whose offspring distribution is in the domain of attraction of an $\alpha$-stable law with $\alpha \ino (1, 2]$. We denote by $\cR_n$ the range of the BRW, i.e., ~the set of all sites in $\bT$ visited by the BRW. Under a moment assumption for $\nu$, we prove that if we view $\cR_n$ as a random subtree of $\bT$ equipped with its graph distance $d_{\mathtt{gr}}$ and with its occupation measure $\ttm^{_{(n)}}_{{\mathtt{occ}}}$ then there exists a scaling sequence $s_n \! \to \! \infty$ such that conditionally given the environment $\bT$, the measured metric space $(\cR_n, s_n^{-1}d_{\mathtt{gr}} , \frac{_1}{^n}\ttm^{_{(n)}}_{{\mathtt{occ}}} )$ weakly converges in the Gromov--Hausdorff--Prokhorov sense to a random measured compact real tree introduced by Curien, Le Gall \& Miermont in \cite{CuLGMi13} called the Brownian cactus with $\alpha$-stable branching mechanism. This work extends in random environment the result from D., K., Lin \& Torri \cite{DuKhLiTo22} which deals with the case where $\bT$ is a regular tree.