Negative large deviations of the front velocity of $N$-particle branching Brownian motion

TL;DR

This paper analyzes the probability of rare negative deviations in the front velocity of a one-dimensional $N$-particle branching Brownian motion system, revealing a phase transition and universal behaviors through macroscopic fluctuation theory.

Contribution

It introduces a detailed analysis of negative large deviations in front velocity using macroscopic fluctuation theory, identifying a phase transition and universal rate functions.

Findings

Rate function matches universal Fisher-KPP class for small deviations.

Existence of a critical velocity $c_*$ with a second-order phase transition.

For large negative deviations, the rate function reaches a simple bound.

Abstract

We study negative large deviations of the long-time empirical front velocity of the center of mass of the one-sided $N$-BBM ($N$-particle branching Brownian motion) system in one dimension. Employing the macroscopic fluctuation theory, we study the probability that $c$ is smaller than the limiting front velocity $c_0$, predicted by the deterministic theory, or even becomes negative. To this end we determine the optimal path of the system, conditioned on the specified $c$. We show that for $c_0-c\ll c_0$ the properly defined rate function $s(c)$, coincides, up to a non-universal numerical factor, with the universal rate functions for front models belonging to the Fisher-Kolmogorov-Petrovsky-Piscounov universality class. For sufficiently large negative values of $c$, $s(c)$ approaches a simple bound, obtained under the assumption that the branching is completely suppressed during the whole time. Remarkably, for all $c\leq c_*$, where $c_*<0$ is a critical value that we find numerically, the rate function $s(c)$ is \emph{equal} to the simple bound. At the critical point $c=c_*$ the character of the optimal path changes, and the rate function exhibits a dynamical phase transition of second order.