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.

## You are here

# Random Connection Models and Mean-Field Critical Exponents

Matthew Dickson (UBC)

Monday, February 12, 2024 - 2:30pm to 3:20pm

THO 119