An infrared bound for the marked random connection model

We investigate a spatial random graph model whose vertices are given as a marked Poisson process on $R^d$. Edges are inserted between any pair of points independently with probability depending on the Euclidean distance of the two endpoints and their marks. Upon variation of the Poisson density, a percolation phase transition occurs under mild conditions: for low density there are finite connected components only, while for large density there is an infinite component almost surely. Our interest is on the transition between the low- and high-density phase, where the system is critical. We prove that if dimension is high enough and the mark distribution satisfies certain conditions, then an infrared bound for the critical connection function is valid. This implies the triangle condition, thus indicating mean-field behaviour. We achieve this result through combining the recently established lace expansion for Poisson processes with spectral estimates in Hilbert spaces. Based on joint work with Matthew Dickson.