We investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a nontrivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. Based on joint work with Peter Gracar and Lukas Luechtrath.

We study the random connection model (without weights) in a regime where a percolation phase transition occurs: for low edge density there are finite connected components only, while for large density there is an infinite component a.s. Our interest is on the transition between the low- and high-density regime. We prove an infrared bound for the critical connection function if the dimension is sufficiently large or if the pair connectivity has sufficiently slow decay; this allows a detailed description of this transition. To this end, we devise the lace expansion for Poisson processes. Based on joint work with Remco van der Hofstad, Günter Last, and Kilian Matzke.