Aaron Smith

The result of an NSERC USRA at University of Toronto. We get a pretty good description of what happens to a family of conformally flat manifolds under the Yamabe flow.

Answers this question in the affirmative, using a pretty explicit non-Markovian coupling as the main tool. Includes a discussion of the relationship to coupling from the past .

This paper contains analyses of two Gibbs samplers on high-dimensional Euclidean space. The first generalizes the paper above as well as two papers on Gibbs samplers written by Dana Randall and Peter Winkler. The second demonstrates the use of some path-coupling techniques that don't work on discrete spaces.

This paper was inspired by "Random doubly stochastic tridiagonal matrices" by Diaconis and Wood. They study mixing time and lack of cutoff for random birth and death chains with uniform stationary distributions. I use coupling techniques rather than their more precise algebraic techniques to extend these analyses to a wide variety of stationary distributions, including families which do exhibit cutoff with probability 1. I also discuss other measures on random chains, geometries other than the path, and analyze a new perfect sampling algorithm for random birth and death chains.

I discuss a random walk on upper-triangular matrices over finite fields with p elements. The paper focuses on the evolution of the top right-hand entry of the matrix, and confirms a conjecture about its mixing as $p$ goes to infinity.