Large deviations for the empirical distribution in the branching random walk

Abstract

We consider the branching random walk (Zn)n≥0 on ℝ where the underlying motion is of a simple random walk and branching is at least binary and at most decaying exponentially in law. For a large class of measurable sets A ⊆ ℝ, it is well known that (formula presented) almost surely as n → ∞, where Zn is the particles empirical distribution at generation n and ν is the standard Gaussian measure on ℝ. We therefore analyze the rate at which (formula presented) and (formula presented) go to zero for any ϵ. We show that the decay is doubly exponential in either n or √n, depending on A and ϵ and find the leading coefficient in the top exponent. To the best of our knowledge, this is the first time such large deviation probabilities are treated in this model.