发表机构
University of Illinois Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对ℓ固定、n→∞的ℓ-扩展线图首通渗流，通过枢轴节点分解方法，得出通行时间Tₙ的大数定律、高斯CLT或非高斯稳定律收敛结果，还证明了最小路径边数的相关极限定理，揭示了Tₙ波动的分布相变。
AI 中文摘要
我们研究ℓ-扩展线图上的首通渗流，其中每个顶点i∈{0,1,…,n}与所有距离不超过ℓ的顶点相连。本文聚焦微观 regime，即ℓ固定而n→∞的情况，为这些边赋予独立非负权重。我们得到从0到n的通行时间Tₙ的大数定律和精确波动结果：若权重分布具有有限方差，或指数大于2/ℓ_c的重尾分布（其中ℓ_c=ℓ(ℓ+1)/2），则Tₙ满足√n缩放的高斯中心极限定理（CLT）；相反，对于指数低于该阈值的更重尾分布，我们证明经适当中心化和缩放后的Tₙ收敛到非高斯稳定律。我们还对最小路径的边数证明了大数定律和CLT，关键工具是枢轴节点分解，测地线可被分割为独立同分布（i.i.d.）块，形成更新结构。我们的结果将经典一维CLT扩展至包含有限范围连通性和重尾的情况，揭示了Tₙ波动中的新分布相变。
英文摘要
We study first-passage percolation on the $\ell$-spread-out line graph, where each vertex $i\in\{0,1,\dots,n\}$ is connected to all others at distance at most $\ell$. Here, we focus on the microscopic regime, with $\ell$ fixed as $n\to\infty$. Independent nonnegative weights are assigned to these edges. We obtain a law of large numbers and precise fluctuation results for the passage time $T_n$ from $0$ to $n$. If the weight distribution has finite variance or a heavy tail with exponent above $2/\ell_c$ where $\ell_c=\ell(\ell+1)/2$, then $T_n$ satisfies a Gaussian CLT with $\sqrt{n}$ scaling. In contrast, for heavier-tailed distributions, with index below the threshold, we show that $T_n$, appropriately centered and scaled, converges to a non-Gaussian stable law. We also prove an LLN and CLT for the number of edges in the minimizing path. The key tool is a pivot-node decomposition; the geodesic can be segmented into i.i.d. blocks, leading to a renewal structure. Our results extend the classical one-dimensional CLT to include finite-range connectivity and heavy tails, revealing a new distributional phase transition in the fluctuations of $T_n$.
