The contact process on Scale-Free Percolation

We consider the contact process on scale-free percolation, a spatial random graph model where the degree distribution of the vertices follows a power law with exponent β. We study the extinction time τGn of the contact process on the graph restricted to a d-dimensional box of volume n, starting from full occupancy. In the regime β∈(2,3), where the degrees have finite mean but infinite variance and the graph exhibits the ultra-small world behaviour, we adapt the techniques of [Linker et al., 2021] to show that τGn is exponential in n. Our main contribution, though, deals with the case β≥3, where the degrees have finite variance and the graph is small-world. We prove that also in this case τGn grows exponentially, at least up to a logarithmic correction reflecting the sparser graph structure. The proof requires the generalization of a result from [Mountford et al., 2016] and combines a multi-scale analysis of the graph, the study of the chemical distance between vertices and percolation arguments.