9:30--10:15 Terence Tao: Brown measure and the circular law Abstract: The circular law asserts that the normalised spectral distribution of
a large square random matrix with iid entries of mean zero and
variance one, is asymptotically given by the uniform measure on the
unit disk. A lot of effort over many decades has gone into the proof
of this law. On the other hand, it is relatively easy (though still
not trivial) to show that the free probability limit of such matrices
(assuming all moments are finite) has the Brown measure predicted by
the circular law.
As is well known, Brown measure does not always correspond to spectral
measure in general. Nevertheless, by using tools from the most recent
work on the circular law by Van Vu and myself, one can show that Brown
measure and asymptotic spectral measure in fact coincide for any
random matrix ensemble with independent (but not necessarily
identically distributed) entries, even when the mean and variance of
the entries are allowed to vary (but obeying some uniform boundedness
conditions). We discuss this result in this talk. |
11:00--11:45 Charles Bordenave: Spectrum of non-hermitian heavy tailed random matrices Abstract: Consider a $n \times n$ matrix with independent and identically distributed
entries. If the variance is finite, the circular law Theorem asserts
that the empirical spectral distribution converges to the uniform
distribution on the unit complex disc. This theorem was recently
proved by Tao and Vu after important breakthroughs of Girko, Bai and
others. In this talk, we will consider the infinite variance case. We
will state the convergence of the empirical spectral measure and give
some properties of the limiting distribution. This is a joint work
with Djalil Chafaï and Pietro Caputo. |
Lunch break |
1:30--2:15 Alice Guionnet: The single ring theorem Abstract: will talk about a recent work with O. Zeitouni and
M. Krishnapur.
We study the empirical measure $L_{A_n}$
of the eigenvalues of non-normal square matrices
of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar distributed
on the unitary group and $D_n$ real diagonal. We show that when the
empirical measure of the eigenvalues of $D_n$ converges, and $D_n$
satisfies some technical conditions, $L_{A_n}$
converges
towards a rotationally invariant measure on the complex plan whose
support is a single ring and which was first described
by Haagerup and Larsen. In particular, we
provide a complete
proof of Feinberg-Zee single ring theorem. |
3:00--3:45 Nizar Demni: Ultraspehrical type generating functions and generalized Cauchy Stieltjes transforms Abstract: we characterize generating functions of ultraspherical type via differential equations. As by product, we relate generalized Cauchy-Stieltjes transforms of some Beta distributions and powers of ordinary Cauchy-Stieltjes transforms. We also recover another characterization of the free Meixner family. Then, we focus on a deformation of the Wigner distribution and compute its moments and free cumulants. Finally, some similarities with generalized Gamma convolutions are shown. |
4:00--4:45 Mylene Maida: Central limit theorem for unitary Brownian motion Abstract: This is joint work with Thierry L\'evy (CNRS and Universit\'e de Gen\`eve).
P. Diaconis and S. Evans established a central limit theorem for traces of functions
of uniformly distributed unitary matrices and the covariance involved the $H^{1/2}$-scalar product.
We consider the case of unitary Brownian motion, for which P. Biane described the limiting process
as the dimension goes to infinity, the so-called free multiplicative Brownian motion and we looked in this work
at the fluctuations.
I will discuss both combinatorial and probabilistic approaches of this problem, try to explain the form of the covariance we get (via a martingale approach) and if time permits, show that this covariance converges as time grows to the $H^{1/2}$-scalar product pointed out by Diaconis and Evans. |