Large Random Matrices: Lectures on Macroscopic Asymptotics

École d'Été de Probabilités de Saint-Flour XXXVI ¿ 2006

Authors:

ISBN: 978-3-540-69896-8 (Print) 978-3-540-69897-5 (Online)

Table of contents (21 chapters)

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  1. Front Matter

    Pages I-XII

  2. No Access

    Book Chapter

    Pages 1-4

    Introduction

  3. Wigner matrices and moments estimates

    1. Front Matter

      Pages 5-5

    2. No Access

      Book Chapter

      Pages 7-28

      Wigner’s theorem

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      Book Chapter

      Pages 29-40

      Wigner's matrices; more moments estimates

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      Book Chapter

      Pages 41-46

      Words in several independent Wigner matrices

  4. Wigner matrices and concentration inequalities

    1. Front Matter

      Pages 47-48

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      Book Chapter

      Pages 49-57

      Concentration inequalities and logarithmic Sobolev inequalities

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      Book Chapter

      Pages 59-64

      Generalizations

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      Book Chapter

      Pages 65-87

      Concentration inequalities for random matrices

  5. Matrix models

    1. Front Matter

      Pages 89-91

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      Book Chapter

      Pages 93-107

      Maps and Gaussian calculus

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      Book Chapter

      Pages 109-118

      First-order expansion

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      Book Chapter

      Pages 121-145

      Second-order expansion for the free energy

  6. Eigenvalues of Gaussian Wigner matrices and large deviations

    1. Front Matter

      Pages 147-148

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      Book Chapter

      Pages 149-158

      Large deviations for the law of the spectral measure of Gaussian Wigner's matrices

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      Book Chapter

      Pages 159-162

      Large Deviations of the Maximum Eigenvalue

  7. Stochastic calculus

    1. Front Matter

      Pages 166-166

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      Book Chapter

      Pages 167-181

      Stochastic analysis for random matrices

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      Book Chapter

      Pages 183-209

      Large deviation principle for the law of the spectral measure of shifted Wigner matrices

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      Book Chapter

      Pages 211-216

      Asymptotics of Harish-Chandra-Itzykson-Zuber integrals and of Schur polynomials

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      Book Chapter

      Pages 217-224

      Asymptotics of some matrix integrals

  8. Free probability

    1. Front Matter

      Pages 266-266

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      Book Chapter

      Pages 227-230

      Free probability setting

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      Book Chapter

      Pages 231-244

      Freeness

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      Book Chapter

      Pages 245-259

      Free entropy

  9. Appendix

    1. Front Matter

      Pages 261-261

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      Book Chapter

      Pages 263-265

      Basics of matrices

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