

Working seminar on 

integrable systems and
random matrices 




Spring 2007 


Thursdays 3:30  4:30pm,
Courant Institute WWH Room 513 

Organizers: Jinho Baik
and Percy Deift (baik@cims.nyu.edu, deift@cims.nyu.edu) 


Date 
Speaker 
Title and Abstract 

18Jan 
Percy Deift 
On the Pfaffian 


(CIMS) 
The
solution of many problems in mathematics and mathematical 



physics
can be expressed in terms of the Pfaffian of a skew symmetric matrix. 



This
talk is an introduction to some basic properties of the Pfaffian. 



Various applications will also be discussed. 

8Feb 
Rowan Killip 
A road less travelled 


(UCLA) 
I will
outline an approach to random matrix theory first 



pursued
by H. Trotter. The route appears to
lead in interesting 



directions, but also has plenty of potholes. 

15Feb 
Peter Miller 
The
Semiclassical Modified Nonlinear Schrodinger 


(U. Michigan) 
Equation: Facts and Artifacts 



I will discuss some recent work (joint with J. DiFranco) on
semiclassical Cauchy 



problems
for an integrable perturbation of the focusing nonlinear
Schr\"odinger 



equation.
The perturbation is singular in the sense of inversescattering and also in
the 



more
practical sense that the modified equation admits solutions with
surprisingly 



different properties than the unmodified equation.
"Facts and Artifacts" is a reference to 



a similarlytitled paper by E. V. Doktorov,
whose lecture on the subject in Edinburgh in 



2004 originally piqued our interest. 

22Feb 
Rowan Killip 
Eigenvalue statistics for CMV mstrices 

starts at 
(UCLA) 
Per request, I will discuss some of the technicalities 

4:00pm 

of my recent joint work on eigenvalue statistics for CMV
matrices 

1Mar 
Alexei Onatski 
How Large Random Matrices Help Macroeconomists 


(Columbia U. 
The central bank processes a huge amount of information before
it decides on the level of 


Economics dept.) 
the
shortterm interest rate. However, formal economic and statistical
models 



employed to inform this decision rely only on few economic
indicators. I will describe 



undesirable consequences of this contrast, discuss modern
econometric techniques 



which
reduce the contrast, relate these techniques to recent research on large
random 



matrices, describe some new
results, and point out directions for future research. 

8Mar 
Robert Buckingham 
Asymptotics of the TracyWidom distribution functions 


(U. Michigan) 
We
find the constant term of the asymptotic expansions of the TracyWidom
distribution 



functions as the distribution parameter approaches minus
infinity. The TracyWidom distribution 



functions
are written in terms of an integrals of a Painlev\'e II function up to
positive infinity. The 



constants are computed by deriving alternate formulas for the
distribution functions starting from 



minus infinity. The
results for the GOE and GSE distribution functions are new, while the result 



for
the GUE distribution function provides an alternate proof of the recent work
of Deift, Its, and 



Krasovsky.
The new integral formula also allows the evaluation of the total integral of
the 



HastingsMcLeod
solution of the Painlev\'e II equation. This is joint work with Jinho
Baik and 



Jeff DiFranco. 

15Mar 
No seminar 
Spring Break 

22Mar 
No seminar 


29Mar 
Dmitry Shepelsky 
Asymptotics for the CamassaHolm equation 


(Institutte for Low 
We discuss the longtime asymptotis of solutions of the
CamassaHolm equation 


Temperature Physics 
$$u_tu_{txx}+2u_x+3uu_x=2u_xu_{xx}+uu_{xxx},$$ which is an
integrable 


and Engineering) 
model equation
describing the shallowwater approximation in inviscid hydrodynamics. 



The
method is the adaptation of the nonlinear steepest descent method for
RiemannHilbert 



problems. This is joint work with Anne Boutet de Monvel. 

5Apr 
No seminar 


12Apr 
Lauren Williams 
Tableaux combinatorics for the asymmetric exclusion process 


(Harvard) 
The partially asymmetric exclusion process (PASEP) is an
important model from statistical 



mechanics which describes a
system of interacting particles hoppingleft and right on a one 



dimensional lattice of N
sites. It is partially asymmetric in
the sense that the probability of hopping 



left is q times the probability
of hopping right. Additionally, particles may enter from the left with 



probability alpha and exit to the right with probability beta.
It has been observed that the (unique) 



stationary distribution of the PASEP has remarkable
connections to combinatorics. We will describe 



how in fact the
(normalized) probability of being in a particular state of the PASEP can be
viewed as 



a
certain weight generating function for the permutation tableaux (certain
tableaux that come 



indirectly
from the totally nonnegative part of the Grassmannian) of a fixed
shape. Our first proof is 



algebraic
and uses the matrix ansatz of Derrida et al. Our second proof involves
defining a Markov 



chain  which we call the PT chain  on the set of
permutation tableaux which projects to the PASEP 



in a very strong sense, thus revealing a hidden structure
behind the PASEP. Via the bijection from 



permutation tableaux to permutations, the PT chain can also be
viewed as a Markov chain on the 



symmetric group. Another nice feature of the PT chain is that
it possesses a certain symmetry which 



extends the particlehole symmetry of the PASEP. This is joint work with Sylvie Corteel. 

19Apr 
Sandrine Peche 
The spectral radius of large real symmetric random matrices 


(UC Davis) 
with non symmetrically distributed entries 



We
consider random symmetric matrices H_N, of size NXN, with independent 



entries
above the diagonal. Consider $H_N=\frac{1}{\sqrt N}(Hij)$ where the
H_{ij} 



are bounded random variables which are centered with variance
$\sigma^2$ 



but non symmetrically distributed. It is known that the
spectral measure of such 



random
matrix ensembles converges as $N \to \infty$ to the semicircle law.
Using 



the moment methods developped by A. Soshnikov, we prove that
the spectral 



radius is bounded from above by $ 2 \*\sigma +
o(N^{6/11+\epsilon})$, 



where $\epsilon $ is an arbitrary small positive number. 

26Apr 














