Random Connection Models and Mean-Field Critical Exponents 

Random connection models (RCMs) form a family of random graph
models embedded in R^d. Their vertex set is distributed as a
Poisson point process and edges then form independently with a
probability depending on the relative displacement of the two vertices.
Many of these models exhibit a percolation phase transition, and one may
be interested in how properties such as the expected cluster size and
the percolation probability behave near the critical point. If one were
to cheat and assume some independence that does NOT exist, then such
behaviors would be exactly calculable and obey power laws with given
exponents. If the RCMs actually do satisfy these power law behaviors
with these exponents, they are said to have mean-field critical
exponents. We will give conditions (including a version of the triangle
condition) that show that some RCMs have mean-field critical exponents.
We will also consider RCMs in which the vertices are randomly assigned
marks that affect the probability of edges forming. This is based on
joint work with Alejandro Caicedo and Markus Heydenreich.