Illinois Journal of Mathematics

Markovian loop clusters on graphs

Yves Le Jan and Sophie Lemaire

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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 $d\geq2$ and the complete graph. These examples show the relations between Poissonian ensembles of Markov loops and other models: renewal process, percolation and random graphs.

Article information

Illinois J. Math., Volume 57, Number 2 (2013), 525-558.

First available in Project Euclid: 19 August 2014

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G55: Point processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C40: Connectivity


Le Jan, Yves; Lemaire, Sophie. Markovian loop clusters on graphs. Illinois J. Math. 57 (2013), no. 2, 525--558. doi:10.1215/ijm/1408453593.

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