We are interested in the branching capacity of the range of a random walk in Zᵈ.Schapira [28] has recently obtained precise asymptotics in the case d≥ 6 and has demonstrated a transition at dimension $d=6$. We study the case $d=5$ and prove that the renormalized branching capacity converges in law to the Brownian snake capacity of the range of a Brownian motion. The main step in the proof relies on studying the intersection probability between the range of a critical Branching random walk and that of a random walk, which is of independent interest.
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Bai et al. (2024) studied this question.