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Random doubly stochastic tridiagonal matrices
Author(s) -
Diaconis Persi,
Wood Philip Matchett
Publication year - 2013
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20452
Subject(s) - tridiagonal matrix , mathematics , combinatorics , markov chain , eigenvalues and eigenvectors , distribution (mathematics) , physics , mathematical analysis , quantum mechanics , statistics
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