Branching capacity of a random walk in Z⁵*

We are interested in the branching capacity of the range of a random walk in Z^d. 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.