2016/02/16 by Andreas E. Kyprianou, Kyprianou, Andreas E., Steven W. Pagett +5
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1602.05241
An important property of Kingman's coalescent is that, starting from a state\nwith an infinite number of blocks, over any positive time horizon, it\ntransitions into an almost surely finite number of blocks. This is known as\n`coming down from infinity'. Moreover, of the many different (exchangeable)\nstochastic coalescent models, Kingman's coalescent is the `fastest' to come\ndown from infinity. In this article we study what happens when we counteract\nthis `fastest' coalescent with the action of an extreme form of fragmentation.\nWe augment Kingman's coalescent, where any two blocks merge at rate c>0, with\na fragmentation mechanism where each block fragments at constant rate,\n\λ>0, into it's constituent elements. We prove that there exists a phase\ntransition at \λ=c/2, between regimes where the resulting `fast'\nfragmentation-coalescence process is able to come down from infinity or not. In\nthe case that \λ<c/2 we develop an excursion theory for the fast\nfragmentation-coalescence process out of which a number of interesting\nquantities can be computed explicitly.\n