Lec 1 | 05 Jan |
Introduction Gaussian and Semicircle laws |
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Notes |
Lec 2 | 10 Jan |
Gaussian and semicircle laws, Catalan numbers |
Stanley's exercise
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Lec 3 | 12 Jan |
Convergence of measures. Wigner matrices. ESD of a matrix. Statement of Wigner's semicircle law |
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Lec 4 | 17 Jan |
Method of moments and WSL for expected ESD of GOE matrix |
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Lec 5 | 19 Jan |
Expected ESD of GOE matrix. Connection to maps on surfaces.
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Notes |
Lec 6 | 24 Jan |
Continuity properties of eigenvalues of Hermitian matrices
Rank inequality and Hoffman-Wielandt (von Neumann)
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Lec 7 | 25 Jan |
Method of moments to prove WSL for general Wigner matrices
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Notes |
Lec 8 | 07 Feb |
Stieltjes tranforms, basic properties, inversion formula and continuity theorem
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Lec 9 | 09 Feb |
Wigner's semicircle law by Stieltjes' transform method, under fourth moment assumption
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Lec 10 | 14 Feb |
Stieltjes' transform proof (cont'd). Remarks on fourth moment assumption.
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Notes |
Lec 11 | 16 Feb |
Chatterjee's invariance principle. Proof of Wigner's semicircle law under Pastur's condition
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Lec 12 | 07 Mar |
Tridiagonalization of GUE and GOE matrices.
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Lec 13 | 09 Mar |
Tridiagonal matrices and probability measures on the line.
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Notes |
Lec 14 | 14 Mar |
Eigenvalue density for β-tridiagonal matrices. Statistical mechanics interpretation. Selberg integral.
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Notes |
Lec 15 | 16 Mar |
The special case β=2. Determinantal form of the density. Marginal densities, counting number of points in a subset.
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Lec 16 | 21 Mar |
Circular unitary ensemble. Mean and variance of linear statistics. Asymptotic normality for N(I).
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Lec 17 | 23 Mar |
Fredholm determinants and hole probability in determinantal processes. Asymptotics of gap probability in CUE.
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Lec 18 | 25 Mar |
Hermite polynomials and their basic properties. Integral representations. Christoffel-Darboux formula.
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Notes |
Lec 19 | 28 Mar |
Laplace's method and the saddle point method. Semicircle law, bulk and edge scaling for GUE. |
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Lec 20 | 30 Mar |
Elements of free probability theory. Cumulants in classical probability theory. |
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Notes |
Lec 21 | 06 Apr |
Noncommutative probability spaces.
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Lec 22 | 11 Apr |
Free independence. Examples. Free cumulants.
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Lec 23 | 13 Apr |
Free cumulants and fee independence. Free CLT.
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Lec 24 | 18 Apr |
Relationship to random matrix theory. Voiculescu's theorem on the sum of wo Hermitian random matrices. |
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Subhamay Saha Symmetric band matrices and Wigner's semicircle law |
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The paper |
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Rajesh Sundaresan Random matrices in communication theory |
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Slides |
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Tulasi Ram Reddy Smallest singular value of an i.i.d matrix
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The paper |
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