2017/03/19 by Aser Cortines, Cortines, Aser, Lisa Hartung +3 · 4 citations
Mathematics · #60G15 #60G70 #60J80 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60G15 #msc:60G70 #msc:60J80
paper · pdf · doi:10.48550/arxiv.1703.06529
Changes from previous version: Proofs of random walk estimates exported to the supplement manuscript: "decorated random walk restricted to stay below a curve (supplement material)". Theorem 1.4 (in previous version) exported to the standalone manuscript: "more on the structure of extreme level sets in branching Brownian motion". Current version accepted to Annals of Probability
arxiv created 2019/02/21 · arxiv updated 2019/02/22
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 and shape of those clusters which 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, 2011. The proofs rely on studying the cluster distribution and should carry over to the branching random walk and the two-dimensional discrete Gaussian free field with no conceptual difficulty.