Lower deviation probabilities for level sets of the branching random walk

Given a branching random walk$\{Z_n{\}}_{n\ge 0}$ on $\mathbb{R}$, let $Z_n([y,\mathrm{\infty }))$ be the number of particles located in $[y,\mathrm{\infty })$ at generation $n$. It is known from \cite{Biggins1977} that under some mild conditions, $n^{-1}\log Z_n([\theta x^{\ast }n,\mathrm{\infty }))$ converges a.s. to $\log m-I(\theta x^)$,where$\log m-I(\theta x^)$ is a positive constant. In this work, we investigate its lower deviation, in other words, the convergence rates of $$\mathbb{P}(Z_n([\theta x^{\ast }n,\mathrm{\infty }))<e^{an}),$$ where $a\in [0,\log m-I(\theta x^{\ast }))$. Our results complete those in \cite{Mehmet}, \cite{Helower} and \cite{GWlower}.