Large Components in the Random Connection Model

Abstract

We explore the Marked Random Connection Model (MRCM) in the subcritical and supercritical regimes.
The behavior of Large Components will be used to guide our exploration.

In the subcritical regime we show that large components occupy a vanishing fraction of the observation window. We do this by studying the correlation length, which describes the tail behavior of components, and is an object of interest in its own right.

In the supercritical regime we show that the largest component occupies a strictly positive fraction of the observation window. Our analysis will require various `uniqueness' statements, which ensure that distant clusters are indeed connected. We prove the Grimmett-Marstrand theorem for the MRCM to help us sharpen our uniqueness statements, which we can then employ to prove the desired result about large components.

Date of Award	18 Feb 2026
Original language	English
Awarding Institution	University of Bath
Supervisor	Mathew Penrose (Supervisor) & Cecile Mailler (Supervisor)