AI 中文总结
该研究分析了ℓ-扩展一维周期图上首通渗的两种几何区域（介观、宏观）的渐近行为，确定了临界连通性阈值，实现了从高斯波动到极值波动的范围插值。
AI 中文摘要
我们研究大小为n的ℓ-扩展一维周期图上的首通渗，其中顶点在图距离不超过ℓ时相连。对固定θ>0，我们给边分配独立同分布的非负随机权重，服从威布尔分布ω_e ~ Exp(1)^(1/θ)。本文研究当连通性参数ℓ随n发散时，两个典型顶点间的通行时间T_n及最优路径的跳数在渐近行为上的转变。我们确定了两种根本不同的几何区域：在介观区域（1≪ℓ≪n），最优路径局部模仿空间分支随机游走，但全局仍受一维几何约束；我们建立了以Crump-Mode-Jagers（CMJ）分支随机游走前沿速度为特征的大数定律，证明当ℓ≪n^(1/4)时存在高斯波动的中心极限定理，且期望跳数与空间距离成正比增长。在宏观区域（ℓ≈λn，λ∈(0,1/2)），图变为高度连通的平均场网络；我们证明通行时间坍缩到log n尺度，具有常数阶非高斯波动，明确确定了由两个独立非空间CMJ过程碰撞驱动的极值极限，并建立了跳数的大数定律。最后，我们严格追踪T_n均值阶在两个区域间的转变，证明通行时间在临界连通性阈值ℓ≍n/log n处存在阶转变；我们的结果提供了从空间高斯波动到平均场极值波动的全面确定性范围插值。
英文摘要
We study first-passage percolation on the $\ell$-spread-out one-dimensional cycle of size $n$, where vertices are connected if their graph distance is at most $\ell$. We assign i.i.d.~non-negative random weights from a Weibull distribution $ω_e \sim \mathrm{Exp}(1)^{1/θ}$ to the edges for $θ>0$ fixed. This paper investigates the transition in the asymptotic behavior of the passage time $T_n$ between two typical vertices and the hop-count of the optimal path as the connectivity parameter $\ell$ diverges with $n$. We identify two fundamentally distinct geometric regimes. In the mesoscopic regime ($1 \ll \ell \ll n$), the optimal path locally mimics a spatial branching random walk but remains globally constrained to a one-dimensional geometry. We establish a law of large numbers characterized by the front speed of a Crump--Mode--Jagers branching random walk, prove a central limit theorem with Gaussian fluctuations when $\ell\ll n^{1/4}$, and show that the expected hop-count grows proportionally with the spatial distance. In the macroscopic regime ($\ell \approx λn$ for $λ\in (0,1/2)$), the graph becomes a highly connected mean-field network. We prove that the passage time collapses to a $\log n$ scale with constant order non-Gaussian fluctuations, explicitly determining the extreme-value limit driven by the collision of two independent non-spatial CMJ processes. We establish a law of large numbers for the hop-count. Finally, we rigorously trace the transition in the order of the mean of $T_n$ between these two regimes, demonstrating an order transition for the passage time across the critical connectivity threshold $\ell \asymp n/\log n$. Our results provide a comprehensive deterministic-range interpolation from spatial Gaussian fluctuations to mean-field extreme-value fluctuations.
