Phase transition in continuum percolation
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. In this minicourse, we outline an expansion technique that allows identifying connectivity properties at and near the transition point under suitable conditions. We discuss the notion of mean-field behaviour in this context, and derive critical exponents that characterise the phase transition. We also draw the connection to related models and open problems.