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### Manjunath Krishnapur

Department of Mathematics, Indian Institute of Science, Bangalore 560 012

### Random matrix theory (Spring 2011)

Mon-Wed 11:00 - 12:30, in LH I

Some generalities: This is a special topics course on random matrix theory. RMT is a huge subject with many themes, of which we shall try to cover only an essential few. Some useful references are the books by Anderson, Guionnet and Zeitouni, by Peter Forrester, by Bai and Silverstein and the old classic by Mehta. Of late many experts have taught courses on the subject and have lecture notes or other material posted on their websites. Some of the online resources I know are by Terence Tao, by Benedek Valko and by Fraydoun Rezakhanlou. The difference in material you see in these links is an indicator of how huge the field has grown already.

I will be posting my own notes and they are highly recommended too! If you find the above references overwhelming, just stick to the book by Anderson, Guionnet and Zeitouni which is available online. Playing with large random matrices on matlab is highly encouraged.

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