AI 中文总结
研究汉明图上分支随机游走覆盖时间的渐近性，关注慢分支情形。通过结合多种技术方法，得到\\(b>2\\)和\\(b = 2\\)时覆盖时间的精确渐近公式，改进了线性阶估计，反映了未覆盖区域几何结构。
AI 中文摘要
我们证明了在汉明图\\(\{0,1,\dots,b - 1\}^d\\)上连续时间分支随机游走的覆盖时间\\(\tau_{\mathrm{cov}}(d)\\)当\\(d\to\infty\\)时的紧密渐近性。我们关注慢分支情形，粒子移动速率为1且分支速率为\\(\lambda\in(0,1)\\)。对于\\(b>2\\)，我们表明\\(\tau_{\mathrm{cov}}(d)=x_\star d+\lambda^{-1}\log d+O_\mathbb{P}(1)\\)；对于\\(b = 2\\)，\\(\tau_{\mathrm{cov}}(d)=x_\star d+\chi^{-1}\log\log d+O_\mathbb{P}(1)\\)。结果改进了先前已知的线性阶估计。二分法反映了最后未覆盖区域的几何结构。证明结合了经典的脊柱测度变换技术、多对少估计、系谱的多尺度分解以及早期种群的加权鞅分析。
英文摘要
We prove tight asymptotics of the cover time $τ_{\mathrm{cov}}(d)$ of a continuous-time branching random walk on the Hamming graph $\{0,1,\dots,b-1\}^d$, as $d\to\infty$. We focus on the slow-branching regime, where particles move at rate one and branch at rate $λ\in(0,1)$. For $b>2$, we show that $τ_{\mathrm{cov}}(d)=x_\star d+λ^{-1}\log d+O_{\mathbb P}(1)$. For $b=2$, we show that $τ_{\mathrm{cov}}(d)=x_\star d+χ^{-1}\log\log d+O_{\mathbb P}(1)$. Here, $x_\star$ and $χ$ are explicit positive constants depending only on $b$ and $λ$. Our results sharpen previously known linear-order estimates. The dichotomy reflects the geometry of the last uncovered region: for $b>2$, there are exponentially many antipodes, whereas the binary hypercube has a unique antipode and its neighbors govern the final coverage. Our proofs combine classic spine change of measure techniques and many-to-few estimates with a multiscale decomposition of the genealogy and a weighted martingale analysis of the early population.
Comments50 pages, 8 figures
