Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment

Abstract

We study inhomogeneous long-range first-passage percolation on $\mathbb{Z}^d$ with edge passage times $\lVert x-y\rVert^\alpha\omega_{xy}/(V_xV_y)$, where vertex weights have polynomial upper-tail exponent $\gamma$ and edge noises have polynomial lower-tail exponent $\theta$ 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 $q_{\rm hub}=d/\gamma$ and $q_{\rm edge}=d/\theta$. 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 $\alpha<q_{\rm hub}\vee q_{\rm edge}$, while $T_n=\Theta_{\mathbb{P}}(1)$ when $q_{\rm hub}\vee q_{\rm edge}<\alpha<2q_{\rm hub}$. In the edge-dominated intermediate regime, $T_n=O_{\mathbb{P}}((\log n)^{\Delta_{\rm III}+\varepsilon})$, where $\Delta_{\rm III}=\log 2/\log(2q_{\rm edge}/\alpha)$. The two power-law regimes satisfy $T_n=O_{\mathbb{P}}(n^{\alpha-2q_{\rm hub}+\varepsilon})$ and $T_n=O_{\mathbb{P}}(n^{\alpha-2q_{\rm edge}+\varepsilon})$, 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.