Year: 2026
Type: preprint
Abstract: We study continuous-time supercritical branching random walks in space-inhomogeneous random environment on the Hamming cube $\{0,1,\dots,b-1\}^d$, where the reproduction laws at each site are i.i.d. sampled. This serves as an idealized model for RNA sequence evolution with mutation--selection balance. The reproduction and mutation events are decoupled, and we assume that the reproduction law has essential supremum strictly below the deterministic mutation rate. Our main results provide tight asymptotics of the first-passage times for the model, uniformly in the origin--target distance $1\le m\le d$ as $d\to\infty$, conditional upon survival. We prove both quenched tightness in environment probability and annealed tightness around a deterministic center. Moreover, we identify the leading order of the deterministic center and develop an expansion in the sparse-distance regime $m=o(d)$. Our proof technique derives quantitative approximations of the model using an independent-visit variant, where revisits still resample the environment. As an application, we show that on a macroscopic scale, increasing the reproduction law in convex order decreases the first-passage times, and increasing the mutation rate increases the first-passage times for the sparse regime.
Source: arXiv (Cornell University)
Authors: Blanchet, Jose, Zhang, Zhenyuan
Institutions: Stanford University, Massachusetts Institute of Technology
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Collaborative Research: CIF: Medium: Statistical and Algorithmic Foundations of Distributionally Robust Policy Learning, Collaborative Research: FET: Medium: Quantum Monte Carlo Speed Ups for Multilevel Computations and Other Statistical Algorithms, Collaborative Research: AMPS: Rare Events in Power Systems: Novel Mathematics, Statistics and Algorithms.
OpenAlex ID: W7220669196