2017/03/28 by Michel Pain, Pain, Michel · 1 citation
Mathematics · Physics and Astronomy · #60F05 #60F17 #60J80 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.1703.09792
openalex publication_date 2017/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure νn,β on the nth generation, which is also the polymer measure on a disordered tree with inverse temperature β. The convergence of the partition function Wn,β, after rescaling, towards a nontrivial limit has been proved by A"ıdékon and Shi in the critical case β= 1 and by Madaule when β>1. We study here the near-critical case, where βn → 1, and prove the convergence of Wn,βn, after rescaling, towards a constant multiple of the limit of the derivative martingale. Moreover, trajectories of particles chosen according to the Gibbs measure νn,β have been studied by Madaule in the critical case, with convergence towards the Brownian meander, and by Chen, Madaule and Mallein 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.