Consistent Minimal Displacement of Branching Random Walks

TL;DR

This paper investigates the asymptotic behavior of minimal displacement in branching random walks on regular trees, establishing a precise convergence rate for a related minimal maximum displacement, thus answering a previously open question.

Contribution

It proves the almost sure convergence of a scaled minimal maximum displacement in branching random walks, providing an explicit constant and resolving an open problem posed by Hu and Shi.

Findings

$L_n/n^{1/3}$ converges almost surely to a constant $l_0$

Established the asymptotic rate of minimal displacement in branching random walks

Provided an explicit value for the convergence constant

Abstract

Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in \mathbb{T}}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $\Lambda(\cdot)$. Let $m_n$ denote the minimum of the variables $S_v$ over all vertices at the $n$th generation, denoted by $\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a constant, which for convenience may be taken to be 0. With $\bar S_v=\max\{S_w:{\rm $w$ is on the geodesic connecting the root to $v$}\}$, define $L_n=\min_{v\in \mathbb{D}_n} \bar S_v$. We prove that $L_n/n^{1/3}$ converges almost surely to an explicit constant $l_0$. This answers a question of Hu and Shi.