Markovian loop clusters on graphs

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 $d\ge 2$ and the complete graph. These examples show the relations between Poissonian ensembles of Markov loops and other models: renewal process, percolation and random graphs.

29 pages, accepted to the Illinois Journal of Mathematics

Cited by in corpus (2)

• Equality of critical parameters for percolation of Gaussian free field level-sets
• A multiplicative coalescent with asynchronous multiple mergers