Flocking under Fast and Large Jumps: Stability, Chaos, and Traveling Waves

We study a model for flocking given by a n-particle system under which each particle jumps forward by a random amount, independently sampled from a given distribution, with rate given by a non-increasing function w of its signed distance from the system center of mass. This model was introduced in Bal\'azs et. al. (2014) and some of its properties were studied for the case when w is bounded. In the current work we are interested in the setting where w is unbounded, and this feature results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We characterize the large n limit (the so-called `fluid limit') of the empirical measure process associated with the system and prove a propagation of chaos result. Next, for the centered n-particle system, by constructing suitable Lyapunov functions, we establish existence and uniqueness of stationary distributions and study their tail properties. In the special case where w is an exponential function and is an exponential distribution, by establishing that all stationary solutions of the McKean-Vlasov equation must be the unique fixed point of the equation, we prove a propagation of chaos result at t= and establish convergence of the particle system, starting from stationarity, in the large n limit, to a traveling wave solution of the McKean-Vlasov equation. The proof of this result may be of interest for other interacting particle systems where convexity properties or functional inequalities generally used for establishing such a result are not available. Our work answers several open problems posed in Bal\'azs et. al.