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.