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The distance between the two BBM leaders

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Published 16 February 2022 • © 2022 IOP Publishing Ltd & London Mathematical Society
, , Citation Julien Berestycki et al 2022 Nonlinearity 35 1558 DOI 10.1088/1361-6544/ac4a8e

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Author e-mails

julien.berestycki@stats.ox.ac.uk

eric.brunet@ens.fr

grahamca@stanford.edu

leonid@ie.technion.ac.il

Jean-Michel.Roquejoffre@math.univ-toulouse.fr

ryzhik@stanford.edu

Author affiliations

1 Department of Statistics and Magdalen College, University of Oxford, United Kingdom

2 Laboratoire de Physique de l'École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France

3 Department of Mathematics, Stanford University, Stanford, CA 94350, United States of America

4 Faculty of Industrial Engineering and Management, Technion, Technion City, Haifa 3200003, Israel

5 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

Author notes

6 Author to whom any correspondence should be addressed.

7 Recommended by Dr Konstantin M Khanin.

ORCID iDs

Dates

  1. Received 29 November 2020
  2. Revised 22 December 2021
  3. Accepted 12 January 2022
  4. Published

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0951-7715/35/4/1558

Abstract

We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit d12 of this random variable can be expressed in terms of PDEs related to the Fisher–KPP equation. We use such a representation to determine the sharp asymptotics of $\mathbb{P}({d}_{12} > a)$ as a → +∞. These tail asymptotics were previously known to 'exponential order;' we discover an algebraic correction to this behavior.

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10.1088/1361-6544/ac4a8e