Markovian loop clusters on graphs

Yves Le Jan (1) , Sophie Lemaire (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Poisson point process of loops; random walk on graphs; coalescent process; percolation; connectivity; renewal process

Domaines: Probabilités [math.PR]

Yves Le Jan, Sophie Lemaire. Markovian loop clusters on graphs. 2013. ⟨hal-00747583v3⟩