Large deviation probabilities for the range of a d-dimensional supercritical branching random walk

Let $\{Z_n{\}}_{n\ge 0}$ be a $d$-dimensional supercritical branching random walk started from the origin. Write $Z_n(S)$ for the number of particles located in a set $S\subset {\mathbb{R}}^d$ at time $n$. Denote by $R_n:=\inf \{\rho :Z_i(\{\mathrm{\mid }x\mathrm{\mid }\ge \rho \})=0,\mathrm{\forall }0\le i\le n\}$ the range of $\{Z_n{\}}_{n\ge 0}$ before time $n$. In this work, we show that under some mild conditions $R_n\mathrm{/}n$ converges in probability to some positive constant $x^{\ast }$ as $n\to \mathrm{\infty }$. Furthermore, we study its corresponding lower and upper deviation probabilities, i.e. the decay rates of $$\mathbb{P}(R_n\le xn)\text{ for }x\in (0,x^{\ast });\mathbb{P}(R_n\ge xn)\text{ for }x\in (x^{\ast },\mathrm{\infty })$$ as $n\to \mathrm{\infty }$. As a by-product, we confirm a conjecture of Engländer \cite{Englander04}.