Processes on Random Geometric Graphs

Competing growth on lattices and graphs

Competing first passage percolation describes the growth of two competing infections on an underlying graph structure. It was first studied on the Z^d-lattice. The main question is if the infection types can grow to occupy infinite parts of the lattice simultaneously, the conjecture being that the answer is yes if and only if the infections grow with the same intensity. Recently, the model has been analyzed on more heterogeneous graph structures, where the degrees of the vertices can have an arbitrary distribution. In this case, it turns out that also the degree distribution plays a role in determining the outcome of the competition. I will give a survey of existing results, both on Z^d and on heterogeneous graphs, and describe open problems. I will also describe a spatial version of the underlying heterogeneous graphs.

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.