Student Probability and Related Fields Seminar

Are you sitting comfortably? Then I'll tell you the story behind Persi Diaconis and Jason Fulman's paper "Foulkes characters, Eulerian idempotents and an amazing matrix", and my attempts to generalise it. It concerns how the descent set of a deck of cards changes as one shuffles.

Comparison is a powerful strategy for the analysis of Markov chains. To study a chain of interest, a related but simpler or more symmetric chain is analyzed completely, and the results transferred to the original chain. This theory has been developed by Diaconis, Saloff-Coste, and others for chains on the same state space. However, in many examples, a natural `symmetrized' walk takes place on a larger state space. I discuss how extensions can be used to compare walks on different state spaces, with some simple illustrations. I'll then go over a more difficult example, comparing simple random walk on derangements to a related simple random walk on permutations. This talk is a mixture of current analyses, future projects, and joint work with John Jiang.

The original Kac random walk only prescribes conservation of energy of the particle system. If one imposes linear momentum conservation, the system degenerates into the random transposition shuffle walk on S_n. In order to have a continuous model, one has to put the particles in higher dimensions > 1. Here I discuss some previous work on these extended models, and derive some new master equation analogous to the walk on the Lie group SO(n). A strange phenomenon occurs in dimension 2. I will also discuss a few related models arising from quantum computing. Finally if time permits, some simulation results on the Wasserstein mixing time of the original Kac walk will be presented.