## ANNOUNCEMENTS

Registration for Spring term opens Thursday, February 23, 2016.

This page was last updated: .

## COURSE DESCRIPTION

This course will provide an introduction to random matrix theory. We will first discuss relevant concentration inequalities and the moment method. Subsequently we will investigate the matrix norm of a random matrix and provide two proofs of Wigner's semi-circle law. After an introduction to free probability, we will focus on Gaussian ensembles and the edge of the spectrum.

## TOPICS COVERED

- Large deviations and concentration of measure
- The operator norm of a random matrix
- Dyck words and Catalan numbers
- Bai--Yin theorem
- Wigner's semi-circe law
- A proof using the moment method
- A proof using the Stieltjes transform

- Free probability
- Gaussian ensembles

## TEXTBOOKS

[1] T. Tao, 'Topics in Random Matrix Theory', Graduate Studies in Mathematics, 132. American Mathematical Society, Providence, RI, 2012. x+282 pp. ISBN: 978-0-8218-7430-1

[2] G. W. Anderson, A. Guionnet and O. Zeitouni, 'An introduction to random matrices', Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. xiv+492 pp. ISBN: 978-0-521-19452-5

## LECTURE NOTES

Date | Description | Reference |
---|---|---|

04/04/16 | Course overview, Universality, Large deviations, Moment method | [1] pp. 55-63 |

04/06/16 | Moment generating function, Truncation method, Weak/Strong law of
large numbers, Talagrand's concentration inequality |
[1] pp. 63-73 |

04/11/16 | Proof of Talagrand's concentration inequality, Distance to
subspaces, Random Matrix ensembles, Operator norm |
[1] pp. 73-75, 78 [1] pp. 105-107 |

04/13/16 | Epsilon net argument, Concentration of measure, 4th moment method | [1] pp. 107-110 [1] pp. 114-117 |

04/18/16 | k-th moment method, Leading order term, Dyck words | [1] pp. 117-122 |

04/20/16 | Weak Bai--Yin theorem (lower/upper bound), Catalan numbers | [1] pp. 122-130 |

04/25/16 | Strong Bai--Yin theorem, Empirical spectral distribution, modes of convergence | [1] pp. 130-135 |

04/27/16 | Wigner's semicircular law, Reductions, Proof of convergence in
expectation using the moment method |
[1] pp. 135-140 [2] pp. 10-15 |

05/02/16 | ||

05/04/16 | ||

05/09/16 | ||

05/11/16 | ||

05/16/16 | ||

05/18/16 | ||

05/23/16 | ||

05/25/16 | ||

05/30/16 | ||

06/01/16 | ||

06/06/16 (undergraduates only) | ||

06/08/18 (undergraduates only) |

## EXAMS

The examination will take place in the form of a student talk on a related topic. A list of possible topics will be provided but suggestions are welcome.

## PRESENTATION TOPICS (to be expanded)

Fom the textbooks:

- Log-Sobolev inequality: pp. 75-77 [1]
- Linbdeberg swapping trick: Section 2.2.4 [1]
- Stein's method of proving CLT: Section 2.2.6 [1]
- Predecessor comparison: Section 2.2.7 [1]
- A symmetrisation argument: Section 2.3.2 [1]
- Dyson Brownian motion and the Steiltjes transform: Section 2.4.4 [1]

Papers:

- N. Alon, M. Krivelevich, V. Vu,
*On the concentration of eigenavalues of random symmetric matrices*, Israel J. Math.**131**(2002), 259-267 - A. Guionnet, O. Zeitouni,
*Concentration of spectral measure for large matrices*, Electron. Comm. Probab.**5**(2000), 119-136