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MA
2010
Springer

The Dirichlet Markov Ensemble

13 years 2 months ago
The Dirichlet Markov Ensemble
We equip the polytope of n × n Markov matrices with the normalized trace of the Lebesgue measure of Rn2 . This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n, . . . , 1/n). We show that if M is such a random matrix, then the empirical distribution built from the singular values of √ n M tends as n → ∞ to a Wigner quarter–circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of √ n M tends as n → ∞ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of M is of order 1 − 1/ √ n when n is large. AMS 2000 Mathematical Subject Classification: 15A52; 15A51; 15A42; 60F15; 62H99.
Djalil Chafaï
Added 29 Jan 2011
Updated 29 Jan 2011
Type Journal
Year 2010
Where MA
Authors Djalil Chafaï
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