Random walks and hyperplane arrangements
Kenneth S. Brown and Persi Diaconis
Source: Ann. Probab. Volume 26, Number 4
(1998), 1813-1854.
Abstract
Let $\mathscr{C}$ be the set of chambers of a real hyperplane arrangement. We study a random walk on $\mathscr{C}$ introduced by Bidigare, Hanlon and Rockmore. This includes various shuffling schemes used in computer science, biology and card games. It also includes random walks on zonotopes and zonotopal tilings. We find the stationary distributions of these Markov chains, give good bounds on the rate of convergence to stationarity, and prove that the transition matrices are diagonalizable. The results are extended to oriented matroids.
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Keywords: RAndom walk; Markov chain; hyperplane arrangement; zonotope; eigenvalues; diagonalizable matrix; oriented matroid
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1022855884
Mathematical Reviews number (MathSciNet): MR1675083
Digital Object Identifier: doi:10.1214/aop/1022855884
Zentralblatt MATH identifier: 0938.60064
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