Markovian loop clusters on the complete graph and coagulation equations
arXiv:1308.4100
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.
version 3: 34 pages, 1 figure, results on the phase transition added