Abstract:
This paper offers a personal review of some things we've learned about rates of convergence of Markov chains to their stationary distributions. The main topic is ways of speeding up diffusive behavior. It also points to open problems and how much more there is to do.

Abstract:
We study a reversible one-dimensional spin system with Bernoulli(p) stationary distribution, in which a site can flip only if the site to its left is in state +1. Such models have been used as simple exemplars of systems exhibiting slow relaxation. We give fairly sharp estimates of the spectral gap as p decreases to zero. The method uses Poincare comparison with a long-range process which is analyzed by probabilistic methods (coupling, supermartingales).

Abstract:
We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory into more classical probability.

Abstract:
This paper develops projection pursuit for discrete data using the discrete Radon transform. Discrete projection pursuit is presented as an exploratory method for finding informative low dimensional views of data such as binary vectors, rankings, phylogenetic trees or graphs. We show that for most data sets, most projections are close to uniform. Thus, informative summaries are ones deviating from uniformity. Syllabic data from several of Plato's great works is used to illustrate the methods. Along with some basic distribution theory, an automated procedure for computing informative projections is introduced.

Abstract:
The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new two-parameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on permutations and the Ewens sampling formula are special cases. The Markov chain is a version of the auxiliary variables algorithm of statistical physics. Properties of the Macdonald polynomials allow a sharp analysis of the running time. In natural cases, a bounded number of steps suffice for arbitrarily large k.

Abstract:
Consider the barycentric subdivision which cuts a given triangle along its medians to produce six new triangles. Uniformly choosing one of them and iterating this procedure gives rise to a Markov chain. We show that almost surely, the triangles forming this chain become flatter and flatter in the sense that their isoperimetric values goes to infinity with time. Nevertheless, if the triangles are renormalized through a similitude to have their longest edge equal to $[0,1]\subset\CC$ (with 0 also adjacent to the shortest edge), their aspect does not converge and we identify the limit set of the opposite vertex with the segment [0,1/2]. In addition we prove that the largest angle converges to $\pi$ in probability. Our approach is probabilistic and these results are deduced from the investigation of a limit iterated random function Markov chain living on the segment [0,1/2]. The stationary distribution of this limit chain is particularly important in our study. In an appendix we present related numerical simulations (not included in the version submitted for publication).

Abstract:
This article gives a rigorous analysis of the fluctuations of the Bose-Einstein condensate for a system of non-interacting bosons in an arbitrary potential, assuming that the system is governed by the canonical ensemble. As a result of the analysis, we are able to tell the order of fluctuations of the condensate fraction as well as its limiting distribution upon proper centering and scaling. This yields interesting results. For example, for a system of $n$ bosons in a 3D harmonic trap near the transition temperature, the order of fluctuations of the condensate fraction is $n^{-1/2}$ and the limiting distribution is normal, whereas for the 3D uniform Bose gas, the order of fluctuations is $n^{-1/3}$ and the limiting distribution is an explicit non-normal distribution. For a 2D harmonic trap, the order of fluctuations is $n^{-1/2}(\log n)^{1/2}$, which is larger than $n^{-1/2}$ but the limiting distribution is still normal. All of these results come as easy consequences of a general theorem.

Abstract:
The number of ``carries'' when $n$ random integers are added forms a Markov chain [23]. We show that this Markov chain has the same transition matrix as the descent process when a deck of $n$ cards is repeatedly riffle shuffled. This gives new results for the statistics of carries and shuffling.

Abstract:
The "carries" when n random numbers are added base b form a Markov chain with an "amazing" transition matrix determined by Holte. This same Markov chain occurs in following the number of descents or rising sequences when n cards are repeatedly riffle shuffled. We give generating and symmetric function proofs and determine the rate of convergence of this Markov chain to stationarity. Similar results are given for type B shuffles. We also develop connections with Gaussian autoregressive processes and the Veronese mapping of commutative algebra.