The Annals of Probability

The structure of extreme level sets in branching Brownian motion

Aser Cortines, Lisa Hartung, and Oren Louidor

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Abstract

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.

Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2257-2302.

Dates
Received: April 2017
Revised: August 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1562205709

Digital Object Identifier
doi:10.1214/18-AOP1308

Mathematical Reviews number (MathSciNet)
MR3980921

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G70: Extreme value theory; extremal processes
Secondary: 60G15: Gaussian processes

Keywords
Branching Brownian motion extreme values cluster processes

Citation

Cortines, Aser; Hartung, Lisa; Louidor, Oren. The structure of extreme level sets in branching Brownian motion. Ann. Probab. 47 (2019), no. 4, 2257--2302. doi:10.1214/18-AOP1308. https://projecteuclid.org/euclid.aop/1562205709


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Supplemental materials

  • Decorated random walk restricted to stay below a curve. Proofs for the random walk statements in Section 2.1.