Random doubly stochastic tridiagonal matrices

Abstract

Let \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally in probability problems as birth and death chains with a uniform stationary distribution. We study ‘typical’ matrices T∈ \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n chosen uniformly at random in the set \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n. A simple algorithm is presented to allow direct sampling from the uniform distribution on \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n. Using this algorithm, the elements above the diagonal in T are shown to form a Markov chain. For large n, the limiting Markov chain is reversible and explicitly diagonalizable with transformed Jacobi polynomials as eigenfunctions. These results are used to study the limiting behavior of such typical birth and death chains, including their eigenvalues and mixing times. The results on a uniform random tridiagonal doubly stochastic matrices are related to the distribution of alternating permutations chosen uniformly at random.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 403–437, 2013

The cutoff phenomenon for random birth and death chains

“…Our primary result, the statement of which is slightly technical, is in Section 6. This generalizes the main result of [9] proving the lack of cutoff for random birth and death chains with uniform stationary distribution to many other families of birth and death chains using a probabilistic comparison technique. We begin with some notation.…”

“…In Section 2.2 of [9], Diaconis and Wood propose several ways to sample from the set A π . They include an exact algorithm, which seemed to be slow in practice, and a Markov chain based algorithm without any rigorous running time bounds, which seemed to be quick in practice.…”

“…They include an exact algorithm, which seemed to be slow in practice, and a Markov chain based algorithm without any rigorous running time bounds, which seemed to be quick in practice. This section describes an algorithm closely related to that used in [9], and provides a proof of its efficiency. Although the algorithm is quite efficient, it is not trivial to implement well.…”

“…Although the algorithm is quite efficient, it is not trivial to implement well. Its analysis is included here to answer a question posed in [9], and also because many of the lemmas proved in its analysis are useful for proving the main theorems about cutoff in this paper. For a practical discussion about implementing this sampler, a very different analysis resulting in a novel perfect sampling algorithm, and an O(n log(n) 2 ) bound on the running time of the original algorithm, see [27].…”

“…These matrices are the transition kernels of birth and death (BD) chains with uniform stationary distribution, and [9] uses detailed knowledge of the set A ( 1 n , 1 n ,..., 1 n ) to obtain information about 'typical' birth and death chain with uniform stationary distribution. In this paper, we extend many of their results to birth and death chains with general stationary distribution.…”