Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment
Abstract
We study inhomogeneous long-range first-passage percolation on ℤd with edge passage times lVert x−yrVertαωxy/(VxVy), 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 qhub=d/γ and qedge=d/θ. For Tn=T(0,⌈ nx⌉), we prove upper bounds of the conjectured order in every regime and matching lower bounds in phases I and II. Specifically, Tn=0 a.s. when α<qhub∨ qedge, while Tn=Θℙ(1) when qhub∨ qedge<α<2qhub. In the edge-dominated intermediate regime, Tn=Oℙ((log n)ΔIII+ε), where ΔIII=log 2/log(2qedge/α). The two power-law regimes satisfy Tn=Oℙ(nα−2qhub+ε) and Tn=Oℙ(nα−2qedge+ε), and the linear regime satisfies Tn=Oℙ(n). All upper bounds are constructive, based on hub-chain and binary edge-bridge multiscale constructions.
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BibTeX
@misc{chatterjee2026inhomogeneous,
title = {{Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment}},
author = {Shirshendu Chatterjee and Partha S. Dey and Daecheol Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20422},
eprint = {2609.20422},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20422},
}
RIS
TY - UNPB
TI - Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment
AU - Shirshendu Chatterjee
AU - Partha S. Dey
AU - Daecheol Kim
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20422
UR - https://arxiv.org/abs/2609.20422
AB - We study inhomogeneous long-range first-passage percolation on ℤ^d with edge passage times lVert x−yrVert^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_hub=d/γ and q_edge=d/θ. For T_n=T(0,⌈ nx⌉), 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_hub∨ q_edge, while T_n=Θ_ℙ(1) when q_hub∨ q_edge<α<2q_hub. In the edge-dominated intermediate regime, T_n=O_ℙ((log n)^Δ_III+ε), where Δ_III=log 2/log(2q_edge/α). The two power-law regimes satisfy T_n=O_ℙ(n^α−2q_hub+ε) and T_n=O_ℙ(n^α−2q_edge+ε), and the linear regime satisfies T_n=O_ℙ(n). All upper bounds are constructive, based on hub-chain and binary edge-bridge multiscale constructions.
ER -