Lower Deviation Probabilities for Level Sets of the Branching Random Walk

Abstract

Given a supercritical branching random walk $$\{Z_n\}_{n\ge 0}$$ on $${\mathbb {R}}$$ , let $$Z_n(A)$$ be the number of particles located in a set $$A\subset {\mathbb {R}}$$ at generation n. It is known from Biggins (J Appl Probab 14:630–636, 1977) that under some mild conditions, for $$\theta \in [0,1)$$ , $$n^{-1}\log Z_n([\theta x^* n,\infty ))$$ converges almost surely to $$\log \left( {\mathbb {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*)$$ as $$n\rightarrow \infty $$ , where $$x^*$$ is the speed of the maximal position of $$\{Z_n\}_{n\ge 0}$$ and $$I(\cdot )$$ 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 $$\begin{aligned} {\mathbb {P}}\left( Z_n([\theta x^* n,\infty ))<e^{an}\right) \end{aligned}$$ as $$n\rightarrow \infty $$ , where $$a\in [0,\log \left( {\mathbb {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*))$$ . 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).