Date: 2018-11-02

Time: 15:30-16:30

Location: BURN 1104

Abstract:

The rigorous mathematical treatment of random fragmentation-coalescent models in the literature is difficult to find, and perhaps for good reason. We examine two different types of random fragmentation-coalescent models which produce somewhat unexpected results.

The first concerns an agent-based model in which, with a rate that depends on the configuration of the system, agents coalesce into clusters that also fragment into their individual constituent membership. We consider the large-scale, long-term behaviour of this system in a similar spirit to recent use of such models to characterise the evolution of terrorist cells. Under appropriate assumptions we find an unusual behaviour; the system displays stabilisation with clusters that only contain an odd number of individuals.

Our second random fragmentation-coalescent model is described from the outset as an infinite exchangeable system of agents. We introduce a variant of Kingman’s Coalescent, which is Markov process on the space of exchangeable partitions of the natural numbers, in which blocks of the partition can fragment into their constituent singletons. We ask the simple question: “Does this model make sense when it begins with an infinite number of blocks?”. In other words we address the notion of the fragmentation-coalescent “coming down from infinity”. Again, we find an unusual behaviour; depending on a counter-intuitive parameter regime, the system may or may not be able to come down from infinity.

This is joint work based on two papers with Steven Pagett, Tim Rogers and Jason Schweinsberg.

Speaker

Andreas Kyprianou is a Professor in the Department of Mathematical Sciences at the University of Bath. His research interests include Branching Processes, Branching Diffusions and Superprocesses. Random Walks, Brownian motion, Levy processes and Self-similar Markov processes, Monte-Carlo simulation of stochastic processes

Organized by the McGill Statistics Group

Seminar website: https://mcgillstat.github.io/