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4 - Some generalities

Published online by Cambridge University Press:  03 February 2011

Greg W. Anderson
Affiliation:
University of Minnesota
Alice Guionnet
Affiliation:
Ecole Normale Supérieure, Lyon
Ofer Zeitouni
Affiliation:
Weizmann Institute/University of Minnesota
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Summary

In this chapter, we introduce several tools useful in the study of matrix ensembles beyond GUE, GOE and Wigner matrices. We begin by setting up in Section 4.1 a general framework for the derivation of joint distribution of eigenvalues in matrix ensembles and then we use it to derive joint distribution results for several classical ensembles, namely, the GOE/GUE/GSE, the Laguerre ensembles (corresponding to Gaussian Wishart matrices), the Jacobi ensembles (corresponding to random projectors) and the unitary ensembles (corresponding to random matrices uniformly distributed in classical compact Lie groups). In Section 4.2, we study a class of point processes that are determinantal; the eigenvalues of the GUE, as well as those for the unitary ensembles, fall within this class. We derive a representation for determinantal processes and deduce from it a CLT for the number of eigenvalues in an interval, as well as ergodic consequences. In Section 4.3, we analyze time-dependent random matrices, where the entries are replaced by Brownian motions. The introduction of Brownian motion allows us to use the powerful theory of Ito integration. Generalizations of the Wigner law, CLTs, and large deviations are discussed. We then present in Section 4.4 a discussion of concentration inequalities and their applications to random matrices, substantially extending Section 2.3. Concentration results for matrices with independent entries, as well as for matrices distributed according to Haar measure on compact groups, are discussed.

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Publisher: Cambridge University Press
Print publication year: 2009

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  • Some generalities
  • Greg W. Anderson, University of Minnesota, Alice Guionnet, Ecole Normale Supérieure, Lyon, Ofer Zeitouni, Weizmann Institute/University of Minnesota
  • Book: An Introduction to Random Matrices
  • Online publication: 03 February 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511801334.005
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  • Some generalities
  • Greg W. Anderson, University of Minnesota, Alice Guionnet, Ecole Normale Supérieure, Lyon, Ofer Zeitouni, Weizmann Institute/University of Minnesota
  • Book: An Introduction to Random Matrices
  • Online publication: 03 February 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511801334.005
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Some generalities
  • Greg W. Anderson, University of Minnesota, Alice Guionnet, Ecole Normale Supérieure, Lyon, Ofer Zeitouni, Weizmann Institute/University of Minnesota
  • Book: An Introduction to Random Matrices
  • Online publication: 03 February 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511801334.005
Available formats
×