Cornell Probability Workshop on Random Matrices   June 2nd - 5th 2007

(organized by Alice Guionnet) 

There will be 3 or 4 talks on Sunday and Monday and 2 talks on Tuesday Morning

We will have an unformal reception at the math department after the last talk on Sunday
and a  speaker's Dinner on Monday.

Jinho Baik:  Asymptotics of Tracy-Widom distribution functions
Abstract: The Tracy-Widom distribution functions are the limiting distributions
of the largest eigenvalues in random matrix theory. They also arise in other areas
in combinatorics and probability such as random permutations and random tilings.
The asymptotic expansions of the TW functions at negative infinity have been
known except for the constant term. We discuss the question of evaluating the constant
term. For the GUE case, the constant term was computed by Deift, Its and
Krasovsky, confirming the conjecture of Tracy and Widom.  We present the GOE/GSE case,
and also discuss the question of computing the total  integal of the Painleve II solutions.
This is a joint work with Buckingham and DiFranco.

Sourav Chatterjee:  Fluctuations of eigenvalues and second order Poincare inequalities
Abstract: Linear statistics of eigenvalues in many familiar classes of random matrices
are known to obey gaussian central limit theorems. The proofs of such results are usually
rather difficult, involving hard computations specific to the model in question.
I will present a unified technique for deriving such results via relatively soft arguments.
The approach is based on a notion of `extending the Poincare inequality to the second order' via
Stein's method of normal approximation. Just as ordinary Poincare inequalities give variance bounds,
our second order Poincare inequalities (based on second order partial derivatives) give central limit
theorems. Some examples, complete with total variation error bounds, will be worked out.

Alexander Gamburd:  Averages of characteristic polynomials from classical groups
Abstract: We present  an elementary and self-contained derivation of
formulae for averages of products and ratios of characteristic polynomials
of random matrices from classical groups. Connections with combinatorics and
number theory will also be discussed.

Alice Guionnet:  Enumeration of maps and random matrices
Abstract:  (This talk is based on joint works with E. Maurel Segala and D. Shlyakhtenko)
The relation between random matrices and the enumeration of certain graphs has been used
in physics in many different contexts since the work of 't Hooft and Brezin, Itzykson, Parisi
and Zuber. We shall describe this relation  and show how classical ideas from probability
theory (concentration, diffusion) can be developped to analyse it.

Ken McLaughlin: Random matrices, random matrices with source, asymptotics of
orthogonal polynomials, equilibrium measures and some algebraic curves
Abstract:  The hope is to connect all the words in the title.  We'll introduce random matrices. 
Then we'll describe the use of asymptotics of orthogonal polynomials to study eigenvalue statistics.
We'll repeat this for random matrices with source term.  If there's time we'll describe a key
feature of the asymptotic analysis:  existence and regularity of equilibrium measures,
for random matrices with and without source term.


Sandrine Peche:  Universality results for largest eigenvalues of random matrices
Abstract: we will speak about the moment approach developed by A. Soshnikov to prove
universality results for extreme eigenvalues of large Wigner random matrices. We will also
explain how this approach can be modified to consider more general ensembles of random matrices.

Dimitri Shlyakthenkho:  Free entropy dimension
Abstract. [This talk is in part joint work with A. Guionnet]
Free entropy dimension was introduced by Voiculescu as a kind of fractal dimension
that goes with his definition of free entropy.  It is a number associated to a finitely-generated
algebra of operators with a positive trace.  We describe connections between this number,
L^2 cohomology and free stochastic calculus.  Finally, we work out the properties of the
classical analog (in which free entropy is replaced by classical entropy), which ends up being
a certain fractal dimension of a probability measure on R^n.

Balint  Virag : Hyperbolic Brownian motion and the sine point process
Abstract: The bulk point process limit of general beta matrix ensembles can be
expressed as a simple functional of Brownian motion in the hyperbolic
plane. This is joint work with Benedek Valko.

Harold Widom:  Integral formulas for the asymmetric simple exclusion process
Abstract: Exclusion processes were introduced by Frank Spitzer in the late 1960s.
We discuss a special case, the asymmetric simple exclusion process on the integers.
Each particle waits exponential time, then with probability p it moves one step to the
right if the site is unoccupied, otherwise it stays put; with probability 1-p it moves one step
to the left if the site is unoccupied, otherwise it stays put. We describe how to obtain a
differential equation for the probability of a configuration at time t, given the initial
configuration. Then using the Bethe Ansatz we (this is joint work with Craig A. Tracy)
find an integral formula for the solution. From this one derives formulas for the probability
that a particular particle is at a particular site at time t. For the special case of the totally
asymmetric simple exclusion process (p=1) our formulas reduce to known ones, including one
obtained by Johansson by other means connecting this to a Laguerre random matrix ensemble.

Tentative schedule:

Sunday June 3rd (406 Malott Hall)
10:00-10:55  Registration
11:00-12:00  Harold Widom  Integral formulas for the asymmetric simple exclusion process
12:05-2:40    Lunch
2:45-3:45      Balint  Virag  Hyperbolic Brownian motion and the sine point process
4:00-5:00      Sandrine Peche  Universality results for largest eigenvalues of random matrices
5:05-6:00      Reception (Malott Lounge (5th floor))

Monday June 4th (406 Malott)
9:45-10:45   Ken McLaughlin Random matrices, random matrices with source, asymptotics of
                    orthogonal polynomials, equilibrium measures and some algebraic curves
11:00-12:00 Dimitri Shlyakthenkho  Free entropy dimension
12:05-2:25   Lunch
3:00-4:00     Alexander Gamburd  Averages of characteristic polynomials from classical groups
4:15-5:15     Alice Guionnet  Enumeration of maps and random matrices

Tuesday June 5th (406 Malott)
9:45-10:45   Sourav Chatterjee Fluctuations of eigenvalues and second order Poincare inequalities
11:00-12:00 Jinho Baik  Asymptotics of Tracy-Widom distribution functions