Lower Deviation Probabilities for Level Sets of the Branching Random Walk

Given a supercritical branching random walk { Z n } n ≥ 0 on R , let Z n ( A ) be the number of particles located in a set A ⊂ R at generation n . It is known from Biggins (J Appl Probab 14:630–636, 1977) that under some mild conditions, for θ ∈ [ 0 , 1 ) , n - 1 log Z n ( [ θ x ∗ n , ∞ ) ) converges almost surely to log E [ Z 1 ( R ) ] - I ( θ x ∗ ) as n → ∞ , where x ∗ is the speed of the maximal position of { Z n } n ≥ 0 and I ( · ) is the large deviation rate function of the underlying random walk. In this work, we investigate its lower deviation probabilities, in other words, the convergence rates of P Z n ( [ θ x ∗ n , ∞ ) ) < e an as n → ∞ , where a ∈ [ 0 , log E [ Z 1 ( R ) ] - I ( θ x ∗ ) ) . Our results complete those in Chen and He (Ann Institut Henri Poincare Probab Stat 56:2507–2539, 2020), Gantert and Höfelsauer (Electron Commun Probab 23(34):1–12, 2018) and Öz (Latin Am J Probab Math Stat 17:711–731, 2020).