Markovian loop clusters on the complete graph and coagulation equations

2013/08/19 by Sophie Lemaire, Lemaire, Sophie · 1 citation
Mathematics · #05C80 #60C05 #60J80 #82C31 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:05C80 #msc:60C05 #msc:60J80 #msc:82C31

paper · pdf · doi:10.48550/arxiv.1308.4100

version 3: 34 pages, 1 figure, results on the phase transition added

arxiv created 2014/06/17 · arxiv updated 2014/06/18

Abstract

Poissonian ensembles of Markov loops on a finite graph define a random graph process in which the addition of a loop can merge more than two connected components. We study Markov loops on the complete graph derived from a simple random walk killed at each step with a constant probability. Using a component exploration procedure, we describe the asymptotic distribution of the connected component size of a vertex at a time proportional to the number of vertices, show that the largest component size undergoes a phase transition and establish the coagulation equations associated to this random graph process.

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• A multiplicative coalescent with asynchronous multiple mergers

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• Markovian loop clusters on graphs
• Loop clusters on complete graphs
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