Central limit theorem and large deviations for branching random walks driven by products of random matrices

A branching random walk is a system of particles, where each particle gives birth to new particles of the next generation, which move on $\mathbb R^d$ according to some probability law. In this talk, we consider the new model in which the position of a particle is obtained by the action of a matrix on the position of its parent, where the matrices corresponding to different particles are independent and identically distributed. This permits us to extend significantly the domains of applications of the classical theory of branching random walks where particles move according to an additive random walk. We study the asymptotic properties of the counting measure which counts the number of particles of generation n situated in a given region. A central limit theorem and precise large deviation result will be presented. As an important ingredient of the proof, a sufficient and necessary condition for the non-degeneracy of the limit of the fundamental martingale related to the products of random matrices along branches, will also be presented.

(The talk is based on a joint work with Thi Thuy Bui and Ion Grama.)