# Slower deviations of the branching Brownian motion and of branching random walks

## TL;DR

This paper extends the analysis of large deviation functions for the maximum position in branching processes, revealing phase transitions and complex asymptotic behaviors in more general models beyond branching Brownian motion.

## Contribution

It generalizes previous results to broader branching random walks and uncovers non-trivial power law prefactors in the distribution of the maximum position.

## Key findings

- Large deviation function exhibits a phase transition at a negative velocity.
- Asymptotic distribution of the maximum has a non-trivial power law prefactor.
- Results apply to a wider class of branching random walks.

## Abstract

We have shown recently how to calculate the large deviation function of the position $X_{\max}(t) $ of the right most particle of a branching Brownian motion at time $t$. This large deviation function exhibits a phase transition at a certain negative velocity. Here we extend this result to more general branching random walks and show that the probability distribution of $X_{\max}(t)$ has, asymptotically in time, a prefactor characterized by non trivial power law.