Growth of the extremal and cluster-level sets in branching Brownian motion

Branching Brownian motion (BBM) is a classical probabilistic model that has "log-correlated" behavior. Its limiting extremal process has been derived to be that of a randomly shifted clustered Poisson point process with an exponential intensity (Aidekon-Berestycki-Brunet-Shi; Arguin-Bovier-Kistler). In this talk, I will discuss some recent results on the asymptotic growth of upper-level sets [−v,∞) for the limiting extremal process and the cluster process. We show that the former grows like C⋆Zve^√2v almost surely and the logarithm of the latter grows like √2v minus random fluctuations of order v^{2/3} governed by an explicit law in the limit. The first result improves upon the works of Cortines et al. and Mytnik et al. in which asymptotics are shown in probability, while the second makes rigorous the derivation in the physics literature by Mueller et al. and Le et al. and resolves a conjecture thereof.

This is based on joint work with Oren Louidor and Lisa Hartung.