On the Maximal Displacement of Subcritical Branching Random Walks

TL;DR

This paper investigates the asymptotic behavior of the maximal displacement in subcritical branching random walks, establishing exponential decay rates and phase transitions depending on the displacement speed.

Contribution

It introduces a detailed analysis of the tail probabilities of maximal displacement in subcritical branching random walks, including conditions for phase transitions and extensions to supercritical cases.

Findings

Existence of a critical speed  where tail probabilities transition from bounded to decay to zero.

Tail distribution of the maximum displacement exhibits exponential decay with specific parameters.

Results extend to supercritical branching random walks conditioned on extinction.

Abstract

We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each $n\in\mathbb{N},$ let $M_{n}$ be the rightmost position reached by the branching random walk up to generation $n$. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists $\rho>1$ such that the function \[ g(c,n):=\rho ^{cn} P(M_{n}\geq cn), \quad \mbox{for each }c>0 \mbox{ and } n\in\mathbb{N}, \] satisfies the following properties: there exist $0<\underline{\delta}\leq \overline{\delta} < {\infty}$ such that if $c<\underline{\delta}$, then $$ 0<\liminf_{n\rightarrow\infty} g (c,n)\leq \limsup_{n\rightarrow\infty} g (c,n) {\leq 1}, $$ while if $c>\overline{\delta}$, then \[ \lim_{n\rightarrow\infty} g (c,n)=0. \] Moreover, if the jump distribution has a finite right range $R$, then $\overline{\delta} < R$. If furthermore the jump distribution is "nearly right-continuous", then there exists $\kappa\in (0,1]$ such that $\lim_{n\rightarrow \infty}g(c,n)=\kappa$ for all $c<\underline{\delta}$. We also show that the tail distribution of $M:=\sup_{n\geq 0}M_{n}$, namely, the rightmost position ever reached by the branching random walk, has a similar exponential decay (without the cutoff at $\underline{\delta}$). Finally, by duality, these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior.