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

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 $c\le c_{\ast }$, where $c_{\ast }<0$ is a critical value that we find numerically, the rate function $s(c)$ is \emph{equal} to the simple bound. At the critical point $c=c_{\ast }$ the character of the optimal path changes, and the rate function exhibits a dynamical phase transition of second order.