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

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.