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

Dec 2022

Let $\{Z_n\}_{n\geq 0 }$ be a $d$-dimensional supercritical branching randomwalk started from the origin. Write $Z_n(S)$ for the number of particleslocated in a set $S\subset\mathbb{R}^d$ at time $n$. Denote by$R_n:=\inf\{\rho:Z_i(\{|x|\geq \rho\})=0,\forall~0\leq i\leq n\}$ the range of$\{Z_n\}_{n\geq 0 }$ before time $n$. In this work, we show that under somemild conditions $R_n/n$ converges in probability to some positive constant$x^*$ as $n\to\infty$. Furthermore, we study its corresponding lower and upperdeviation probabilities, i.e. the decay rates of $$\mathbb{P}(R_n\leqxn)~\text{for}~x\in(0,x^*);~\mathbb{P}(R_n\geq xn) ~\text{for}~x\in(x^*,\infty)$$ as $n\to\infty$. As a by-product, we confirm a conjecture ofEngl\"{a}nder \cite{Englander04}.

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