发表机构
Stanford University; Massachusetts Institute of Technology（斯坦福大学; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究 Hamming 立方体上超临界分支随机游走在随机环境中的首达时间，给出淬火与退火紧性渐近，并揭示繁殖律与突变率对首达时间的影响。
AI 中文摘要
我们研究 Hamming 立方体 $\{0,1,\dots,b-1\}^d$ 上空间非齐次随机环境中的连续时间超临界分支随机游走，其中每个位点的繁殖律是独立同分布采样的。该模型作为具有突变-选择平衡的 RNA 序列进化的理想化模型。繁殖和突变事件是解耦的，我们假设繁殖律的本质本质上确界严格低于确定性突变率。我们的主要结果提供了该模型首达时间的紧渐近，在给定存活的条件下，对原点-目标距离 $1\le m\le d$ 一致成立，当 $d\to\infty$ 时。我们证明了环境概率下的淬火紧性和围绕确定性中心的退火紧性。此外，我们确定了确定性中心的领头阶，并在稀疏距离区域 $m=o(d)$ 中发展了展开式。我们的证明技术利用独立访问变体推导模型的定量近似，其中重访仍然重新采样环境。作为应用，我们表明在宏观尺度上，按凸序增加繁殖律会减少首达时间，而增加突变率会在稀疏区域增加首达时间。
英文摘要
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.
Comments88 pages, 4 figures; ongoing work
