On Operators and Mean-Field Percolation in Marked Random Connection Models
Marked Random Connection Models are random graph models. The vertex set is given by a Poisson point process with a given intensity on the product of $\mathbb{R}^d$ and a mark space, and the probability that two vertices share an edge is given independently with a probability that depends on the spatial positions of the two vertices as well as their respective marks. In many such models, a percolation phase transition can be observed at a finite and positive vertex intensity. It is conjectured that various percolation quantities should have critical exponents as the intensity approaches the critical intensity, and that if the spatial dimension $d$ is sufficiently high (conjectured to be typically $d>6$), then these critical exponents should take particular values (called Mean-Field values) associated with analogous critical exponents for (Markovian) branching random walks. This association suggests that operator descriptions that are fruitful for the simpler random walk problems can also be useful for describing percolation problems. Indeed we will show here that certain classes of marked random connection models achieve mean-field critical exponents, using operators to navigate difficulties introduced by the marks. Based on joint work with Alejandro Caicedo (LMU M{\"u}nchen) and Markus Heydenreich (Universit{\"a}t Augsburg).