Markovian loop clusters on graphs

TL;DR

This paper investigates the properties and dynamics of loop clusters generated by Poissonian ensembles of Markov loops on various graphs, revealing connections to models like percolation and random graphs.

Contribution

It introduces a coalescent process framework for Markov loop clusters and analyzes their behavior on different graph structures, linking to existing probabilistic models.

Findings

Loop clusters form a coalescent process on graph vertices.

Connections established between Markov loop ensembles and percolation models.

Behavior of loop clusters varies across different graph types.

Abstract

We study the loop clusters induced by Poissonian ensembles of Markov loops on a finite or countable graph (Markov loops can be viewed as excursions of Markov chains with a random starting point, up to re-rooting). Poissonian ensembles are seen as a Poisson point process of loops indexed by 'time'. The evolution in time of the loop clusters defines a coalescent process on the vertices of the graph. After a description of some general properties of the coalescent process, we address several aspects of the loop clusters defined by a simple random walk killed at a constant rate on three different graphs: the integer number line $\mathbb{Z}$, the integer lattice $\mathbb{Z}^d$ with $d\geq 2$ and the complete graph. These examples show the relations between Poissonian ensembles of Markov loops and other models: renewal process, percolation and random graphs.