Lower Deviation Probabilities for Level Sets of the Branching Random Walk

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 set $$A\subset {\mathbb at generation n. It is known from Biggins (J Appl Probab 14:630–636, 1977) that under some mild conditions, for $$\theta \in [0,1)$$ $$n^{-1}\log Z_n([\theta x^* n,\infty ))$$ converges almost surely to $$\log \left( {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*)$$ as $$n\rightarrow \infty $$ where $$x^*$$ speed maximal position and $$I(\cdot )$$ large deviation rate function underlying walk. In this work, we investigate its lower probabilities, other words, convergence rates $$\begin{aligned} {P}}\left( ))<e^{an}\right) \end{aligned}$$ $$a\in [0,\log x^*))$$ . Our results complete those Chen He (Ann Institut Henri Poincare Stat 56:2507–2539, 2020), Gantert Höfelsauer (Electron Commun 23(34):1–12, 2018) Öz (Latin Am J Math 17:711–731, 2020).