Branching capacity of a random walk in Z5

We are interested in the branching capacity of the range of a random walk in Zd. Schapira [29] 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.