The structure of extreme level sets in branching Brownian motion

We study the structure of extreme level sets of a standard one-dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height of the local maxima whose clusters carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida (J. Stat. Phys. 143 (2011) 420-446). The proofs rely on a careful study of the cluster distribution.