2024/05/27 by Lisa Hartung, Hartung, Lisa, Oren Louidor +3 · 1 citation
Mathematics · #60G15 #60G70 #60J80 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2405.17634
openalex publication_date 2024/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study the limiting extremal and cluster point processes of branching Brownian motion. The former records the heights of all extreme values of the process, while the latter records the relative heights of extreme values in a genealogical neighborhood of order unity around a local maximum thereof. For the extremal point process, we show that the mass of upper level sets [-v, ∞) grows as C_⋆ Z v e√(2) v(1+o(1)) as v → ∞, almost surely, where Z is the limit of the associated derivative martingale and C_⋆ ∈ (0, ∞) is a universal constant. For the cluster point process, we show that the logarithm of the mass of [-v, ∞) grow as √(2)v minus random fluctuations of order v2/3, which are governed by an explicit law in the limit. The first result improves upon the works of Cortines et al. (arXiv:1703.06529) and Mytnik et al. (arXiv:2009.02042) in which asymptotics are shown in probability, while the second makes rigorous the derivation in the physics literature by Mueller et al. (arXiv:1910.06382) and Le et al. (arXiv:2207.07672) and resolves a conjecture thereof.