发表机构
The City College of New York; University of Illinois Urbana-Champaign（纽约市立学院; 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
本文研究随机顶点环境中非齐次长程首达渗流，提出八区域相图猜想，并证明各区域的首达时间上界及部分下界，揭示重顶点枢纽与边噪声的竞争机制。
AI 中文摘要
我们研究$\mathbb{Z}^d$上的非齐次长程首达渗流，其边通过时间为$\lVert x-y\rVert^\alpha\omega_{xy}/(V_xV_y)$，其中顶点权重具有多项式上尾指数$\gamma$，边噪声在零处具有多项式下尾指数$\theta$。该模型在长程首达渗流与无标度渗流之间插值，并展现出可复用重顶点枢纽与成对特定小噪声桥之间的竞争。我们猜想一个八区域相图，由$q_{\rm hub}=d/\gamma$和$q_{\rm edge}=d/\theta$控制，组织为五个增长阶段。对于$T_n=T(0,\lceil nx\rceil)$，我们在每个区域证明猜想阶的上界，并在阶段I和II中证明匹配的下界。具体地，当$\alpha<q_{\rm hub}\vee q_{\rm edge}$时$T_n=0$几乎必然，而当$q_{\rm hub}\vee q_{\rm edge}<\alpha<2q_{\rm hub}$时$T_n=\Theta_{\mathbb{P}}(1)$。在边主导的中间区域，$T_n=O_{\mathbb{P}}((\log n)^{\Delta_{\rm III}+\varepsilon})$，其中$\Delta_{\rm III}=\log 2/\log(2q_{\rm edge}/\alpha)$。两个幂律区域满足$T_n=O_{\mathbb{P}}(n^{\alpha-2q_{\rm hub}+\varepsilon})$和$T_n=O_{\mathbb{P}}(n^{\alpha-2q_{\rm edge}+\varepsilon})$，线性区域满足$T_n=O_{\mathbb{P}}(n)$。所有上界都是构造性的，基于枢纽链和二元边桥多尺度构造。
英文摘要
We study inhomogeneous long-range first-passage percolation on $\mathbb{Z}^d$ with edge passage times $\lVert x-y\rVert^αω_{xy}/(V_xV_y)$, where vertex weights have polynomial upper-tail exponent $γ$ and edge noises have polynomial lower-tail exponent $θ$ at zero. The model interpolates between long-range first-passage percolation and scale-free percolation and exhibits competition between reusable heavy-vertex hubs and pair-specific small-noise bridges. We conjecture an eight-regime phase diagram, organized into five growth phases governed by $q_{\rm hub}=d/γ$ and $q_{\rm edge}=d/θ$. For $T_n=T(0,\lceil nx\rceil)$, we prove upper bounds of the conjectured order in every regime and matching lower bounds in phases I and II. Specifically, $T_n=0$ a.s. when $α<q_{\rm hub}\vee q_{\rm edge}$, while $T_n=Θ_{\mathbb{P}}(1)$ when $q_{\rm hub}\vee q_{\rm edge}<α<2q_{\rm hub}$. In the edge-dominated intermediate regime, $T_n=O_{\mathbb{P}}((\log n)^{Δ_{\rm III}+\varepsilon})$, where $Δ_{\rm III}=\log 2/\log(2q_{\rm edge}/α)$. The two power-law regimes satisfy $T_n=O_{\mathbb{P}}(n^{α-2q_{\rm hub}+\varepsilon})$ and $T_n=O_{\mathbb{P}}(n^{α-2q_{\rm edge}+\varepsilon})$, and the linear regime satisfies $T_n=O_{\mathbb{P}}(n)$. All upper bounds are constructive, based on hub-chain and binary edge-bridge multiscale constructions.
