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

TL;DR

This paper investigates the large deviation probabilities of the range of a supercritical branching random walk in multiple dimensions, establishing convergence results and confirming a prior conjecture about its behavior.

Contribution

It proves the convergence of the scaled range to a positive constant and analyzes the decay rates of large deviation probabilities, confirming a conjecture by Engländer.

Findings

The scaled range R_n/n converges in probability to a positive constant x*.

Derived decay rates for probabilities of the range being significantly smaller or larger than its typical value.

Confirmed Engländer's conjecture on the large deviation behavior of the range.

Abstract

Let $\{Z_n\}_{n\geq 0 }$ 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 $R_n:=\inf\{\rho:Z_i(\{|x|\geq \rho\})=0,\forall~0\leq i\leq n\}$ the range of $\{Z_n\}_{n\geq 0 }$ before time $n$. In this work, we show that under some mild conditions $R_n/n$ converges in probability to some positive constant $x^*$ as $n\to\infty$. Furthermore, we study its corresponding lower and upper deviation probabilities, i.e. the decay rates of $$ \mathbb{P}(R_n\leq xn)~\text{for}~x\in(0,x^*);~\mathbb{P}(R_n\geq xn) ~\text{for}~ x\in(x^*,\infty)$$ as $n\to\infty$. As a by-product, we confirm a conjecture of Engl\"{a}nder \cite{Englander04}.