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Quantum Random Walks and Minors of Hermitian Brownian Motion

Published online by Cambridge University Press:  20 November 2018

François Chapon  and
Manon Defosseux
François Chapon
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6, Paris Cedex 05 email: francois.chapon@upmc.fr
Manon Defosseux
Affiliation:
Laboratoire de Mathématiques Appliquées á Paris 5, Université Paris 5, 75270 Paris Cedex 06 email: manon.defosseux@parisdescartes.fr
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Abstract

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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

46L5360B2014L24quantumrandom walkquantum Markov chaingeneralized casimir operatorsHermitian Brownian motiondiffusionsrandom matricesminor process
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 4 , 01 August 2012 , pp. 805 - 821
Copyright
Copyright © Canadian Mathematical Society 2012

References

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