2025/04/29 by Gabriel Flath, Flath, Gabriel · 1 citation
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2504.20833
openalex publication_date 2025/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a branching Brownian motion (BBM). It is well known \citeBramson1983ConvergenceOS, Lalley1987ACL that the rightmost particle is located near \( mt = √(2) t - (3)/(2√(2)) log t \). Let N(t,x) be the set of particles within distance x from mt, where x = o(t) grows with t. We prove that \(#N(t,x)/π-1/2xexmt/t e-x2/(2t) \) converges in probability to Z_∞, the limit of the so-called derivative martingale, and that, for \( x = O( t1/3) \), the convergence cannot be strengthened to an almost sure result. Moreover, we prove that the asymptotic overlap distribution of two particles sampled uniformly from N(t,x) converges to that of the critical derivative martingale measure. This establishes a universal genealogical picture of the BBM front at sublinear distances from the tip.