The Annals of Applied Probability

Extended convergence of the extremal process of branching Brownian motion

Anton Bovier and Lisa Hartung

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Abstract

We extend the results of Arguin et al. [Probab. Theory Related Fields 157 (2013) 535–574] and Aïdékon et al. [Probab. Theory Related Fields 157 (2013) 405–451] on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the “location” of the particle in the underlying Galton–Watson tree. We show that the limit is a cluster point process on $\mathbb{R}_{+}\times\mathbb{R}$ where each cluster is the atom of a Poisson point process on $\mathbb{R}_{+}\times\mathbb{R}$ with a random intensity measure $Z(dz)\times C\mathrm{e}^{-\sqrt{2}x}\,dx$, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an analogous result for the Gaussian free field by Biskup and Louidor [Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian free field (2016)].

Article information

Source
Ann. Appl. Probab. Volume 27, Number 3 (2017), 1756-1777.

Dates
Received: February 2016
Revised: August 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1500451241

Digital Object Identifier
doi:10.1214/16-AAP1244

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G70: Extreme value theory; extremal processes
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Gaussian processes branching Brownian motion extremal processes cluster processes multiplicative chaos

Citation

Bovier, Anton; Hartung, Lisa. Extended convergence of the extremal process of branching Brownian motion. Ann. Appl. Probab. 27 (2017), no. 3, 1756--1777. doi:10.1214/16-AAP1244. https://projecteuclid.org/euclid.aoap/1500451241.


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References

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