18.338J | Fall 2004 | Graduate
Infinite Random Matrix Theory


LEC # TOPICS Readings
1 Introduction Advances in Random Matrix Theory (PDF)
2 The Hermite Ensemble: Wigner’s Semi-Circle Law Wigner’s Original Paper: Wigner’s Semi-Circle Law

Other Derivations (PDF)

3 The Laguerre Ensemble: Marcenko-Pastur Theorem Silverstein’s Derivation: Paper 1 (PDF), Paper 2 (PDF)

Jonsson’s Paper: Some limit theorems for the eigenvalues of a sample covariance matrix

4 The Jacobi Ensemble: McKay’s Random Graph Theorem McKay’s Paper: The expected eigenvalue distribution of a large regular graph
5 The “Semi-Circular” Element: Central Limit Theorem for Infinite Random Matrices Speicher: Paper 1 (PDF), Paper 2 (PDF)
6 Free Cumulants in Free Probability Speicher: Paper 1 (PDF), Paper 2 (PDF)
7 Free Cumulants in Free Probability (cont.) Speicher: Paper 1 (PDF), Paper 2 (PDF)
8 Non-Crossing Partitions and Free Cumulants Speicher: Paper 1 (PDF), Paper 2 (PDF)
9 Free Cumulants: The Semi-Circular and “Free Poisson” Distributions Speicher: Paper 1 (PDF), Paper 2 (PDF)
10 Additive Free Convolution Speicher: Paper 1 (PDF), Paper 2 (PDF)
11 The R-Transform and the Marcenko-Pastur Theorem Speicher: Paper 1 (PDF), Paper 2 (PDF)
12 Multiplicative Free Convolution Speicher: Paper 1 (PDF), Paper 2 (PDF)
13 The S-Transform Speicher: Paper 1 (PDF), Paper 2 (PDF)
14 Dr. Anna Scaglione Talk  
15 Orthogonal Polynomials and the Classical Matrix Ensembles Ioana’s Paper: Matrix Models for Beta Ensembles
16 Project Progress Presentation  
17 Project Progress Presentation (cont.)  
18 Tracy Widom Distribution Tracy-Widom’s Paper: On the distribution of the largest eigenvalue in principal components analysis

Per’s Paper (PDF)

19 Eigenvalue Spectrum Fluctuations

Johansson’s Paper 1: Shape Fluctuations and Random Matrices (PDF)

Johansson’s Paper 2: On fluctuations of eigenvalues of random Hermitian matrices

Ioana’s Paper: Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics

20 Dr. Roland Speicher Talk  
21 Free Probability and Fluctuations Speicher and Mingo: Paper 1, Paper 2 (PDF)
22 Zonal Polynomials and Random Matrices Stanley’s Paper: Some combinatorial aspects of the spectra of normally distributed random matrices
23 Symmetric Group Representations and Free Probability Phillipe Biane: Paper 1 (PS), Paper 2: Free cummulants, matrix orbits and characters of symmetric groups
24 MOPs Ioana’s Paper: MOPS: Multivariate Orthogonal Polynomials (symbolically)
25 Connections and Open Problems Diaconis’ Paper (PDF)
26 Project Presentations  
27 Project Presentations (cont.)