The investigator will study several problems in probability, most of which concern mixing times for Markov chains. The first problem is to develop tools for using local mixing properties of a Markov chain to understand convergence in the global sense of the mixing time. The investigator hopes to apply these tools to analyze the connection between spatial mixing and temporal mixing in spin sytems. The investigator will also study problems relating to exclusion processes. One example is Aldous's conjecture that in the symmetric exclusion process the spectral gap does not depend on the number of particles. Further problems concern asymmetric exclusion; for some of these models the mixing time is believed to be lower than the corresponding symmetric version but existing techniques cannot verify this.
In recent years a large body of mathematics has been developed relating to finding the mixing time of a Markov chain. A problem that illustrates this body of research is that of determining how many shuffles are necessary to randomize a deck of cards. Here, mathematicians have had great success, finding very precise answers for many models of card shuffling. However, interest in mixing times is not limited to card shuffling since Markov chains are a crucial tool for applications in a wide range of areas. Simulations of Markov chains consume an enormous number of computer cycles each year and are applied in such areas as computer algorithms, statistics and statistical physics.
