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

Abstract: We study inhomogeneous long-range first-passage percolation on $\mathbb{Z}<sup>d$ with edge passage times $\lVert x-y\rVert<sup>αω_{xy}/(V_xV_y)$, 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 $α&lt;q_{\rm hub}\vee q_{\rm edge}$, while $T_n=Θ<em>{\mathbb{P}}(1)$ when $q</em>{\rm hub}\vee q_{\rm edge}&lt;α&lt;2q_{\rm hub}$. In the edge-dominated intermediate regime, $$T_n=O_{\mathbb{P}}((\log n)^{Δ_{\rm III}+\varepsilon})$$, where $Δ<em>{\rm III}=\log 2/\log(2q</em>{\rm edge}/α)$. 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.