Markovian loop clusters on graphs

Yves Le Jan, Sophie Lemaire

Research output: Contribution to journalArticlepeer-review


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 Z, the integer lattice Zd with d ≥ 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.

Original languageEnglish (US)
Pages (from-to)525-558
Number of pages34
JournalIllinois Journal of Mathematics
Issue number2
StatePublished - 2013

ASJC Scopus subject areas

  • General Mathematics


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