Tightness Analysis of First Passage Times of $d$-Dimensional Branching Random Walk

TL;DR

This paper analyzes the asymptotic behavior of the first passage times for a $d$-dimensional supercritical branching random walk, providing tightness results that confirm a recent conjecture.

Contribution

It offers the first precise asymptotics up to O(1) for the first passage time in high dimensions, resolving a conjecture and advancing understanding of branching random walk extremal behavior.

Findings

Established tightness of first passage times as $x o abla$

Provided asymptotic formulas confirming the conjecture in 2024

Analyzed the genealogy and cluster structure near extrema

Abstract

Given a discrete-time non-lattice supercritical branching random walk in $\mathbb{R}^d$, we investigate its first passage time to a shifted unit ball of a distance $x$ from the origin, conditioned upon survival. We provide precise asymptotics up to $O(1)$ (tightness) for the first passage time as a function of $x$ as $x\to\infty$, thus resolving a conjecture in Blanchet--Cai--Mohanty--Zhang (2024). Our proof builds on the previous analysis of Blanchet--Cai--Mohanty--Zhang (2024) and employs a careful multi-scale analysis on the genealogy of particles within a distance of $\asymp \log x$ near extrema of a one-dimensional branching random walk, where the cluster structure plays a crucial role.