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 18-Jan 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. 8-Feb 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. 15-Feb 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 inverse-scattering 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 similarly-titled paper by E. V. Doktorov, whose lecture on the subject in Edinburgh in 2004 originally piqued our interest. 22-Feb 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 1-Mar 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 short-term 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. 8-Mar Robert Buckingham Asymptotics of the Tracy-Widom distribution functions (U. Michigan) We find the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity.  The Tracy-Widom 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 Hastings-McLeod solution of the Painlev\'e II equation.  This is joint work with Jinho Baik and Jeff DiFranco. 15-Mar No seminar Spring Break 22-Mar No seminar 29-Mar Dmitry Shepelsky Asymptotics for the Camassa-Holm equation (Institutte for Low We discuss the long-time asymptotis of solutions of the Camassa-Holm equation Temperature Physics $$u_t-u_{txx}+2u_x+3uu_x=2u_xu_{xx}+uu_{xxx},$$ which is an integrable and Engineering) model equation describing the shallow-water approximation in inviscid hydrodynamics. The method is the adaptation of the nonlinear steepest descent method for Riemann-Hilbert problems. This is joint work with Anne Boutet de Monvel. 5-Apr No seminar 12-Apr 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 non-negative 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 particle-hole symmetry of the PASEP.  This is joint work with Sylvie Corteel. 19-Apr 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 semi-circle 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. 26-Apr