TY - JOUR
T1 - Markovian loop clusters on graphs
AU - Le Jan, Yves
AU - Lemaire, Sophie
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
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U2 - 10.1215/ijm/1408453593
DO - 10.1215/ijm/1408453593
M3 - Article
AN - SCOPUS:84906327351
SN - 0019-2082
VL - 57
SP - 525
EP - 558
JO - Illinois Journal of Mathematics
JF - Illinois Journal of Mathematics
IS - 2
ER -