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

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.

4 pages. Update v2: The proof of Lemma 1 provided in v1 was possibly problematic. A new proof has been provided. Update v3: This preprint is now superseded by arXiv:2411.10333, which actually combines and extends both this preprint and arXiv:2302.05396