On the First Passage Times of Branching Random Walks in $\mathbb R^d$

TL;DR

This work analyzes first passage times for discrete-time branching random walks in $\mathbb{R}^d$ with supercritical genealogy, conditioning on survival. It derives precise asymptotics for the hitting time to a distant unit ball, delivering a linear travel term plus a logarithmic correction, and proves tightness of fluctuations around the median. The authors treat both spherically symmetric and general jump laws, employing frontier-count arguments, random walks in cones, and large-deviation techniques, and extend the framework to a delayed-branching polymer-physics model. The results yield strong laws, density properties of the BRW range, and quantitative connections to shortest-path statistics in polymer networks, with numerical validation across spherical and Gaussian jump scenarios. These findings advance understanding of extremal BRW behavior in higher dimensions and offer practical tools for modeling complex networks where branching and diffusion interact.

Abstract

We study the first passage times of discrete-time branching random walks in ${\mathbb R}^d$ where $d\geq 1$. Here, the genealogy of the particles follows a supercritical Galton-Watson process. We provide asymptotics of the first passage times to a ball of radius one with a distance $x$ from the origin, conditioned upon survival. We provide explicitly the linear dominating term and the logarithmic correction term as a function of $x$. The asymptotics are precise up to an order of $o_{\mathbb P}(\log x)$ for general jump distributions and up to $O_{\mathbb P}(\log\log x)$ for spherically symmetric jumps. A crucial ingredient of both results is the tightness of first passage times. We also discuss an extension of the first passage time analysis to a modified branching random walk model that has been proven to successfully capture shortest path statistics in polymer networks.

Figures (8)

• Figure 1: A two-dimensional visualization of the vector $\mathbf{c}_2$. The range of a (non-spherically symmetric) BRW roughly grows linearly in time (illustrated with the shaded ellipses); at time $\tau_x$, the ellipse is nearly tangent to the target ball $B_x$ (shaded disk). Roughly speaking, the vector $\mathbf{c}_2$ is normal to the tangent line.
• Figure 2: Schematic description of the heuristic for spherically symmetric jumps: around a proportion $x^{-(d-1)/2}$ of the particles that reach roughly $x$ far in the first coordinate (represented by blue dots) lie in the ball $B_x$. The path in red represents (part of) the trajectory that realizes the FPT, whereas the other branches are shown in black.
• Figure 3: Visualization of how random walks in cones are related to estimating first passage times. The first passage time $\tau_x$ is roughly the time when the growing range (illustrated with the shaded ellipses) of the BRW becomes tangent with the target ball $B_x$. In our proof, we will construct moving cones (that are circular centered in the direction $-\mathbf{c}_2$ in the sense of \ref{['eq:circcone']}; the boundaries are indicated by the blue lines) and show that with high probability, the range of the BRW always lies within the moving cones.
• Figure 4: BRW tree representations of polymer networks in the classical and the delayed branching regimes. The red path from $A$ to $A"$ represents the first polymer chain and the blue path from $B$ to $B"$ represents the second chain that cross-links with the first. In the BRW tree genealogy, the particles $A"$, $B$, and $B"$ are the descendants of the particle starting from $A$.
• Figure 5: The numerical 3D BRW obtained $c_1$ compared against the reference calculations across different $p_3$ values for the (a) classical BRW and (b) delayed branching BRW with $p_0=0.004$. The mean of the FPT at each $x$ is computed from $2000$ samples.

Theorems & Definitions (45)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Corollary 5
• Lemma 6: Lemma 2.4 of bramson2016convergence
• Proposition 7
• proof
• Proposition 8
• Remark 1