C O L U M B I A /
C O U R A N T
P R O B A B I L I T Y
S E M I N A R
S E R I E S

Joint Columbia-NYU probability talks, organized by probabilists from both institutions.

####
Friday October 9th, Courant institute, Warren Weaver Hall, 512

- 9.30-10.30am
**Hoi Nguyen** (IAS and Ohio State University).
*Anti concentration of random walks and eigenvalue repulsion of random matrices.*

I will survey recent characterization results on random walks (in both abelian and non-abelian groups) which sticks to a small region unusually long. As an application, we demonstrate a Wegner-type estimate for the number of eigenvalues inside an extremely small interval for Wigner matrices of discrete type.

- 10.30-11am
**Coffee break**
* *

- 11am-12pm
**Nina Snaith** (Bristol).
* Combining random matrix theory and number theory.*

Many years have passed since the initial suggestion by Montgomery (1973) that in an appropriate asymptotic limit the zeros of the Riemann zeta function behave statistically like eigenvalues of random matrices, and the subsequent proposal of Katz and Sarnak (1999) that the same is true of families of more general L-functions. While this limiting behaviour is very informative, even more interesting are the intricacies involved in the approach to this limit, the understanding of which allows us to use random matrix theory in novel ways to shed light on major open questions in number theory.

- 12-1pm
**Ramon van Handel** (Princeton).
* The norm of structured random matrices*.

Understanding the spectral norm of random matrices is a problem of basic interest in several areas of pure and applied mathematics. While the spectral norm of classical random matrix models is well understood, existing methods almost always fail to be sharp in the presence of nontrivial structure. In this talk, I will discuss new bounds on the norm of random matrices with independent entries that are sharp under mild conditions. These bounds shed significant light on the nature of the problem, and make it possible to easily address otherwise nontrivial phenomena such as the phase transition of the spectral edge of random band matrices. I will also discuss some conjectures whose resolution would complete our understanding of the underlying probabilistic mechanisms.