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

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≪c_{0} 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.