# The near-critical Gibbs measure of the branching random walk

## TL;DR

This paper investigates the near-critical behavior of the Gibbs measure in a supercritical branching random walk, establishing convergence results for the partition function and particle trajectories as the inverse temperature approaches criticality.

## Contribution

It extends previous results by analyzing the near-critical regime, proving convergence of the rescaled partition function and describing the limiting behavior of particle trajectories.

## Key findings

- Rescaled partition function converges to a multiple of the derivative martingale limit.
- Particle trajectories under the near-critical Gibbs measure interpolate between Brownian meander and excursion.
- Continuous families of limiting processes are identified from meander to excursion or Brownian motion.

## 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 $n^\text{th}$ 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 A\"{\i}d\'ekon and Shi in the critical case $\beta = 1$ and by Madaule when $\beta >1$. We study here the near-critical case, where $\beta_n \to 1$, 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 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.