Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The near-critical Gibbs measure of the branching random walk

Michel Pain

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Abstract

Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure $\nu_{n,\beta}$ on the $n$th generation, which is also the polymer measure on a disordered tree with inverse temperature $\beta$. The convergence of the partition function $W_{n,\beta}$, after rescaling, towards a nontrivial limit has been proved by Aïdékon and Shi (Ann. Probab. 42 (3) (2014) 959–993) in the critical case $\beta=1$ and by Madaule (J. Theoret. Probab. 30 (1) (2017) 27–63) when $\beta>1$. We study here the near-critical case, where $\beta_{n}\to1$, and prove the convergence of $W_{n,\beta_{n}}$, after rescaling, towards a constant multiple of the limit of the derivative martingale. Moreover, trajectories of particles chosen according to the Gibbs measure $\nu_{n,\beta}$ have been studied by Madaule (Stochastic Process. Appl. 126 (2) (2016) 470–502) in the critical case, with convergence towards the Brownian meander, and by Chen, Madaule and Mallein (On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk (2015) Preprint) in the strong disorder regime, with convergence towards the normalized Brownian excursion. We prove here the convergence for trajectories of particles chosen according to the near-critical Gibbs measure and display continuous families of processes from the meander to the excursion or to the Brownian motion.

Résumé

Considérons une marche aléatoire branchante surcritique réelle dans le cas frontière et la mesure de Gibbs associée $\nu_{n,\beta}$ sur la $n$-ième génération, qui est aussi la mesure de polymère sur un arbre désordonné avec température inverse $\beta$. La convergence de la fonction de partition $W_{n,\beta}$, après renormalisation, vers une limite non-triviale a été démontrée par Aïdékon et Shi (Ann. Probab. 42 (3) (2014) 959–993) dans le cas critique $\beta=1$ et par Madaule (J. Theoret. Probab. 30 (1) (2017) 27–63) pour $\beta>1$. On s’intéresse ici au cas presque-critique, où $\beta_{n}\to1$, et on montre la convergence de $W_{n,\beta_{n}}$, après renormalisation, vers la limite de la martingale dérivée à un facteur multiplicatif près. D’autre part, les trajectoires de particules tirées selon la mesure de Gibbs $\nu_{n,\beta}$ ont été étudiées par Madaule (Stochastic Process. Appl. 126 (2) (2016) 470–502) dans le cas critique, avec convergence vers le méandre brownien, et par Chen, Madaule et Mallein (On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk (2015) Preprint) dans le régime de désordre fort, avec convergence vers l’excursion brownienne. On montre ici la convergence des trajectoires de particules tirées selon la mesure de Gibbs presque-critique et cela fait apparaître une famille continue de processus allant du méandre jusqu’à l’excursion ou jusqu’au mouvement brownien.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1622-1666.

Dates
Received: 8 September 2016
Revised: 31 March 2017
Accepted: 2 July 2017
First available in Project Euclid: 11 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1531296031

Digital Object Identifier
doi:10.1214/17-AIHP850

Mathematical Reviews number (MathSciNet)
MR3825893

Zentralblatt MATH identifier
06976087

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles

Keywords
Branching random walk Additive martingale Trajectories Phase transition

Citation

Pain, Michel. The near-critical Gibbs measure of the branching random walk. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1622--1666. doi:10.1214/17-AIHP850. https://projecteuclid.org/euclid.aihp/1531296031


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