Probability Study Group

This is an informal study group covering topics in probability. Speakers volunteer to present recent papers on topics selected by the group at large. Grad students, postdocs, and faculty are all encouraged to attend.

Details

Random matrices: Getting inside the bulk!

by Van Vu | Yale University
Time: 16:00  (Friday, Mar. 16, 2012)
Location: Stewart Library, Fields Institute, 222 College St
Abstract:
Random matrix theory is a well developed area of mathematics with strong links to various other areas such as mathematical physics, probability, number theory, combinatorics, to mentioned a few. For a very long time, most results in the theory focus on the global distributions of the eigenvalues or the distribution of the eigenvalues at the edge of the spectrum (for example the Tracy-Widom law). Getting information about eigenvalues inside the bulk of the spectrum seemed out of reach and was done in very special cases, such as for matrices with gaussian entries. The situation changed substantially in the last few years, due to the works of Erdos et. al. and Tao and the speaker. In this talk, I am going to present a new method developed by Tao and myself. This method is motivated by the Lindenberg replacement method in probability theory and enabled us to get limiting distribution of every single eigenvalue in the spectrum. This gives a new way to attack many long standing problems. For instance, combining our method with other recent results, one can show the universality of the sine-kernel, confirming an old conjecture of Mehta concerning random hermitian matrices.

Dates in this series

· Monday, Feb. 08, 2010: Liouville Property, Amenability, Discrete Fractals and Automaton Groups (Gidi Amir)
· Monday, Feb. 22, 2010: Scaling Exponents for Directed Polymers in 1+1 Dimensions, Part III (Tom Alberts)
· Monday, Mar. 01, 2010: Scaling exponents for directed polymers in dimension 1+1, part III (Tom Alberts)
· Monday, Apr. 26, 2010: More details of Jeff Schenker's proof of eigenvector localization for band matrices (Ben Rifkind)
· Monday, May. 10, 2010: The Doob h-transform: theory and examples (Alex Bloemendal)
· Monday, May. 17, 2010: Doob's h-transform: a few more examples (Alex Bloemendal)
· Monday, Oct. 04, 2010: Sample Path Properties of Stochastic Flows (Dima Dolgopyat)
· Monday, Oct. 18, 2010: Sample Path Properties of Stochastic Flows, Part II (Dima Dolgopyat)
· Monday, Oct. 25, 2010: Sample Path Properties of Stochastic Flows, Part III (Dima Dolgopyat)
· Friday, Jul. 29, 2011: SLE in multiply connected domains (Gregory F. Lawler)
· Friday, Dec. 02, 2011: A model of migration under constraints (Raoul Normand)
· Friday, Jan. 20, 2012: Local Limit Theorem for Random Walks on Free Groups (Andrew Stewart)
· Friday, Feb. 03, 2012: The Law of the Iterated Logarithm (Eric Hart)
· Friday, Mar. 16, 2012: Random matrices: Getting inside the bulk! (Van Vu)
· Friday, Mar. 30, 2012: A Brief Introduction to Compressed Sensing (Kai Yang)
· Friday, Apr. 13, 2012: The Minimum Cost of a Random Assignment (Mustazee Rahman)
· Tuesday, May. 01, 2012: An Introduction to the Dirichlet Process (Tom Alberts)