Trickle-down growth models, Doob-Martin boundaries, and random matrices
Prof. S. Evans (University of California, Berkeley)
Several Markov chains appearing in applied probability (for example, the random binary search tree and recursive random tree processes) may be viewed as growing connected subsets of a graph that evolve according to the following general dynamics: initially, all vertices of the graph are unoccupied, particles are fed in one-by-one at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each particle comes to rest once it encounters an unoccupied vertex. I will explain how classical tools from Doob-Martin boundary theory may be used to understand the large time asymptotics of such processes. Along the way, we will encounter objects such as Polya urns, Dirichlet random measures, Pitman's generalization of the Ewens sampling formula, and Chinese restaurant processes. If time permits, I will also discuss tools for understanding the asymptotics of the spectra of various random matrices associated with these models.
Regularity and convergence of diffusion processes
Prof. M. Hairer (University of Warwick)
The aim of these lectures will be to provide a number of tools that allow one to determine at which speed (if at all) the law of a diffusion process approaches its stationary distribution. Of particular interest will be cases where this speed is subexponential. At the technical level, the first part of the course will be devoted to an elementary introduction to Malliavin calculus and to a proof of Hörmander's famous "sums of squares" regularity theorem. The second part will be devoted to Lyapunov function techniques, placing emphasis not only on abstract results, but also on techniques of how to construct such functions in practice. Throughout the whole course, an "Ariadne's thread" will be provided by a few apparently simple toy models from statistical mechanics that nevertheless exhibit a surprisingly rich palette of possible behaviours.
Lecture notes: preliminary version.
Participants are encouraged, but by no means obliged, to present their own work in one of the afternoon sessions. Speakers may choose to give an informal introduction to an area of interest, or to present a more detailed account of their research (subject to timetabling constraints). If you would be interested in presenting some work, please indicate this when completing your registration form.