Precise large deviation estimates for the extrema of branching random walks

mercredi 28 janv. 2026, 10:00 → 12:00 Europe/Paris

Johnson (Institut de Mathématiques de Toulouse)

https://rendez-vous.renater.fr/muted_lastN20_user100/Lianghui_Luo_PhD_defence_id_b10a7781e065bbc12e2e1324aa59b73cee04da22648d5c64dc72710225879ef1_0a3c31-939841-6bdda4#config.startWithVideoMuted=true&config.startWithAudioMuted=true

Jury members:
M. Matt ROBERTS, Rapporteur, University of Bath
Mme Manon COSTA, Examinatrice, Université de Toulouse
Mme Xinxin CHEN, Examinatrice, Beijing Normal University
M. Pascal MAILLARD, Examinateur, Université de Toulouse
M. Loïc CHAUMONT, Examinateur, Université d'Angers
M. Bastien MALLEIN, Directeur de thèse, Université de Toulouse

Abstract:

In this thesis, we take interest in the branching random walk, which is a particle system on the real line. Each particle splits and moves at random, independently of the others. In 1976, Biggins proved that the maximal displacement of particles alive at time n grows linearly as n goes to infinity. We obtain large deviation estimates for this quantity. This thesis consists of three chapters.

In the first chapter, we obtain an asymptotic equivalent for the upper large deviation for the maximum of the branching random walk. Besides, we give the limit distribution of the branching random walk seen from its maximum conditionally on this large deviation event. In the second chapter, we obtain an asymptotic equivalent for the upper moderate deviation for the maximum of the branching random walk. This is a fundamental result for the understanding of the fine properties of the invasion of this particle system.

In the final chapter, we consider a two-speed branching random walk, in which particles change their reproduction law at some intermediate time. Thanks to the results obtained in the two previous chapters, we prove the convergence in law of the maximum and the extremal process of the two-speed branching random walk.