Analysis of a Bose-einstein Markov Chain

This paper gives sharp rates of convergence to stationarity for a Markov chain generating Bose-Einstein configurations of n balls in k boxes. The analysis leads to curious identities for the arc sine distribution. 0. On a Personal Note In 1971, as a beginning graduate student at Harvard’s Department of Statistics, I badly wanted to learn “real” probability. Someone told me that the deepest, best book was Paul-Andre Meyers’ “Probability and Potential Theory”. For the next year and a half, three of us ran a reading group on this book. We moved slowly, like ants on a page, without any global understanding but happy to be in the presence of a master. I can’t say I internalized any abstract potential theory but I learned a lot of measure theory and the last chapter (on Choquet Theory) made a big impact on my ability to abstract deFinettis theorem. As the magisterial sequence of books by Dellacherie-Meyer evolved, my familiarity with the original made them welcome and accessible. I only met Paul-Andre Meyer once (at Luminy in 1995). He kindly stayed around after my talk and we spoke for about an hour. I was studying rates of convergence of finite state space Markov chains. He made it clear that, for him, finite state space Markov chains is a trivial subject. Hurt but undaunted, I explained some of our results and methods. He thought about it and said, “I see, yes, those are very hard problems”. The analytic parts of Dirichlet space theory have played an enormous role in my recent work. I am sure that there is much to learn from the abstract theory as well. In the present paper I treat rates of convergence for a simple Markov chain. I am sorry not to have another hour with Paul-Andre Meyer. Perhaps he would say “This piece of our story might help you”. Perhaps one of his students or colleagues can help fill the void. 1. Background The use of Markov chains in Monte Carlo simulations has become a mainstay of scientific computing. There are a bewildering variety of methods for constructing reversible Markov chains with a given stationary distribution. Liu [2001] is a good overview of the present state of the art. Some order has appeared by the realization that many different algorithms are special cases of one general algorithm. Known variously as auxiliary variables (Besag and Green [1993]), data augmentation (Tanner and Wong [1987]), slice sampling (Neale [2003]) or hit and run (Belisle et al. [1991]), these algorithms include the celebrated Swedsen-Wang algorithm of statistical mechanics. They seem to allow “big moves” which suggests rapidly converging chains. There has been very little rigorous work on rates of convergence for any of these algorithms. One spectacular exception are the negative result of Gore and Jerrum (1997) and Borgs et al. [1999] showing that the Swedsen-Wang algorithm does not mix rapidly at the critical temperature. Huber [2000] proves a rapid mixing for Swedsen-Wang suitably far from the critical temperature. The discussion in Neale [2003] contains pointers to a few examples