Markovian loop clusters on the complete graph and coagulation equations

TL;DR

This paper investigates the properties of Markovian loop clusters on the complete graph, revealing phase transitions and deriving coagulation equations for the evolving component sizes in a random graph process.

Contribution

It introduces a detailed analysis of Markov loop clusters on the complete graph, including asymptotic distributions and phase transition phenomena.

Findings

Largest component size exhibits a phase transition

Asymptotic distribution of component sizes characterized

Coagulation equations derived for the process

Abstract

Poissonian ensembles of Markov loops on a finite graph define a random graph process in which the addition of a loop can merge more than two connected components. We study Markov loops on the complete graph derived from a simple random walk killed at each step with a constant probability. Using a component exploration procedure, we describe the asymptotic distribution of the connected component size of a vertex at a time proportional to the number of vertices, show that the largest component size undergoes a phase transition and establish the coagulation equations associated to this random graph process.