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Summer 2013 Markovian loop clusters on graphs
Yves Le Jan, Sophie Lemaire
Illinois J. Math. 57(2): 525-558 (Summer 2013). DOI: 10.1215/ijm/1408453593


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.


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Yves Le Jan. Sophie Lemaire. "Markovian loop clusters on graphs." Illinois J. Math. 57 (2) 525 - 558, Summer 2013.


Published: Summer 2013
First available in Project Euclid: 19 August 2014

zbMATH: 1311.60015
MathSciNet: MR3263044
Digital Object Identifier: 10.1215/ijm/1408453593

Primary: 60C05
Secondary: 05C40 , 60G55 , 60J10 , 60K35

Rights: Copyright © 2013 University of Illinois at Urbana-Champaign


Vol.57 • No. 2 • Summer 2013
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