Up-down Markov chains on partitions and their diffusion analogs

Seminar | February 1 | 3:10-4 p.m. | 1011 Evans Hall

 Soumik Pal, Univ. of Washington

 Department of Statistics

A popular family of models of random partitions is called the Chinese Restaurant Process. We imagine n customers being seated randomly and sequentially at tables of a restaurant according to a fixed stochastic rule. Counting customers by the tables gives us a composition of n. Consider a Markov chain on such compositions where we remove a randomly chosen customer and reseat her. How can one describe the limit of such a Markov chain as n tends to infinity? We will construct such limits as diffusions on partitions of the unit interval. The stationary distributions of such diffusions are given by the complement of the zeros of the Brownian motion or the Brownian bridge. The processes of ranked interval lengths are members of a family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009). Our construction is a piece of a more elaborate diffusion on the space of real trees, stationary with respect to the law of the Brownian Continuum Random Tree, whose existence has been conjectured by David Aldous. Joint work with Noah Forman, Doug Rizzolo, and Matthias Winkel.