Only long edges can erase the subcritical annulus-crossing phase of weight-dependent random connection models

2023/11/07 by Emmanuel Jacob, Jacob, Emmanuel
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2311.04023

Abstract

This short note aims at complementing the results of the recent work arXiv:2302.05396, where Jahnel and Lüchtrath investigate the question of existence of a subcritical percolation phase for the annulus-crossing probabilities in a large class of continuum percolation models, namely the classical or generalized weight-dependent random connection models. Their work relates the absence of a subcritical phase to the occurrence of long edges in the graph, through a somewhat indirect criterium, that stays inconclusive in some models of interest. We provide a more direct and arguably simpler criterium, that is always conclusive when considering classical weight-dependent random connection models.

Related

• Existence of subcritical percolation phases for generalised weight-dependent random connection models
• Subcritical annulus crossing in spatial random graphs
• Supercritical phase of the random connection model
• Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension
• Sharpness of the percolation phase transition for weighted random connection models