Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension

Abstract

We consider inhomogeneous spatial random graphs on the real line. Each vertex carries an i.i.d. weight and edges are drawn such that short edges and edges to vertices with large weights occur with higher probability. This allows the study of models with long-range effects and heavy-tailed degree distributions. We introduce a new coefficient $\delta_\text{eff}$ which quantifies the influence of heavy-tailed degrees on long-range connections. We show that $\delta_\text{eff}<2$ is sufficient for the existence of a supercritical percolation phase in the model and that $\delta_\text{eff}>2$ always implies the absence of percolation. In particular, our results complement those in Gracar et al. (Adv. Appl. Prob., 2021), where sufficient conditions were given for the soft Boolean model and the age-dependent random connection model for both the existence and the absence of a subcritical percolation phase. Our results further provide a criterion for the existence or non-existence of a giant component in large finite graphs.