Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T07:02:49.189Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[ANv M10] Adler, M., Nordenstam, E., and van Moerbeke, P., Consecutive minors for Dyson's Brownian motions. arxiv:1007.0220.Google Scholar
[Bia90] Biane, P., Marches de Bernoulli quantiques. In: Séminaire de Probabilités, XXIV, 1988/89, Lecture Notes in Math., 1426, Springer, Berlin, 1990, pp. 329–344.Google Scholar
[Bia95] Biane, P., Permutation model for semi-circular systems and quantum random walks. Pacific J. Math. 171(1995), no. 2, 373–387.Google Scholar
[Bia06] Biane, P., Le théorème de Pitman, le groupe quantique SUq(2), et une question de P. A. Meyer. In: In memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX, Lecture Notes in Math., 1874, Springer, Berlin, 2006, pp. 61–75.Google Scholar
[Bia08] Biane, P., Introduction to random walks on noncommutative spaces. In: Quantum potential theory, Lecture Notes in Math., 1954, Springer, Berlin, 2008, pp. 61–116.Google Scholar
[Def10] Defosseux, M., Orbit measures, random matrix theory and interlaced determinantal processes. Ann. Inst. Henri Poincaré Probab. Stat. 46(2010), no. 1, 209–249. http://dx.doi.org/10.1214/09-AIHP314 Google Scholar
[Hum78] Humphreys, J. E., Introduction to Lie algebras and representation theory. Second Printing, revised, Graduate Texts in Mathematics, 9, Springer-Verlag, New York-Berlin, 1978.Google Scholar
[KTT92] Klink, W. H. and T. Ton-That, Invariant theory of the block diagonal subgroups of GL(n, C) and generalized Casimir operators. J. Algebra 145(1992), no. 1, 187–203. http://dx.doi.org/10.1016/0021-8693(92)90184-N Google Scholar
[Mey93] Meyer, P.-A., Quantum probability for probabilists. Lecture Notes in Mathematics, 1538, Springer-Verlag, Berlin, 1993.Google Scholar
[Øks03] Øksendal, B., Stochastic differential equations. An introduction with applications. Universitext, Springer-Verlag, Berlin, 2003.Google Scholar
[Žel73] Želobenko, D. P., Compact Lie groups and their representations. Translations of Mathematical Monographs, 40, American Mathematical Society, Providence, RI, 1973.Google Scholar