Math 247a: Introduction to Random Matrix Theory
نویسنده
چکیده
1. Wigner Matrices 2 2. Wigner’s Semicircle Law 4 3. Markov’s Inequality and Convergence of Expectation 6 4. First Proof of Wigner’s Semicircle Law 7 4.1. Convergence of matrix moments in Expectation 7 4.2. Convergence of Matrix Moments in Probability 12 4.3. Weak Convergence 15 5. Removing Moment Assumptions 17 6. Largest Eigenvalue 21 7. The Stiletjes Transform 31 8. The Stieltjes Transform and Convergence of Measures 34 8.1. Vague Convergence 34 8.2. Robustness of the Stieltjes transform 37 8.3. The Stieltjes Transform of an Empirical Eigenvalue Measure 39 9. Gaussian Random Matrices 40 10. Logarithmic Sobolev Inequalities 44 10.1. Heat kernels 47 10.2. Proof of the Gaussian log-Sobolev inequality, Theorem 10.3 51 11. The Herbst Concentration Inequality 54 12. Concentration for Random Matrices 57 12.1. Continuity of Eigenvalues 57 12.2. Concentration of the Empirical Eigenvalue Distribution 59 12.3. Return to Wigner’s Theorem 61 13. Gaussian Wigner Matrices, and the Genus Expansion 61 13.1. GOEn and GUEn 61 13.2. Covariances of GUEn 64 13.3. Wick’s Theorem 65 13.4. The Genus Expansion 67 14. Joint Distribution of Eigenvalues of GOEn and GUEn 70 14.1. Change of Variables on Lie Groups 71 14.2. Change of Variables for the Diagonalization Map 72 14.3. Joint Law of Eigenvalues and Eigenvectors for GOEn and GUEn 75 15. The Vandermonde Determinant, and Hermite Polynomials 77 15.1. Hermite polynomials 78 15.2. The Vandermonde determinant and the Hermite kernel 81
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