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 c0, 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 c0-c≪c0 the properly defined rate function s(c), coincides, up to a nonuniversal 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≤c∗, where c∗<0 is a critical value that we find numerically, the rate function s(c) is 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.