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Quantum Random Walks and Minors of Hermitian Brownian Motion

  Published:2011-09-19
 Printed: Aug 2012
  • François Chapon,
    Laboratoire de probabilités et Modèles Aléatoires, Université Paris 6, Paris Cedex 05
  • Manon Defosseux,
    Laboratoire de Mathématiques Appliquées à Paris 5, Université Paris 5, 75270 Paris Cedex 06
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Abstract

Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.
Keywords: quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process
MSC Classifications: 46L53, 60B20, 14L24 show english descriptions Noncommutative probability and statistics
Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Geometric invariant theory [See also 13A50]
46L53 - Noncommutative probability and statistics
60B20 - Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
14L24 - Geometric invariant theory [See also 13A50]
 

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