ICMP 2003

Integrable systems and random matrix theory

Session organized by P. Deift (New York), M. Jimbo (Tokyo).


J. BaikMichigan
P. DoreyDurham, UK
J.-M. MailletLyon
K. McLaughlinNorth Carolina
A. NakayashikiKyushu
A. OkounkovPrinceton
Jinho Baik
University of Michigan
Ann Arbor

Limiting distribution of random growth models

Many one spatial dimension random growth models are believed to be in the KPZ universality class. Especially the height fluctuation exponent of the models in this class is believed to be 1/3. For a polynuclear growth model, this exponent is proved to be true. Moreover the limiting distribution of the height fluctuation is obtained for this special model. The limiting distributions are more subtle and different symmetry of the model yields different distribution. Some of the distributions are related to the largest eigenvalue of large random Hermitian matrix.

Patrick Dorey

Differential equations and the Bethe ansatz

In this talk I shall review some surprising links which have been discovered in the last few years between the theory of certain ordinary differential equations, and particular integrable lattice models and quantum field theories in two dimensions. An application of this correspondence to a problem in non-Hermitian (PT-symmetric) quantum mechanics will also be discussed.

Jean-Michel Maillet
ENS Lyon

Correlation functions of the XXZ spin-1/2 chain: recent progress

We review recent progress in the computation of correlation functions of the XXZ spin-1/2 chain and their long distance asymptotics. Our method is based on the resolution of the quantum inverse scattering problem in the framework of algebraic Bethe ansatz. It leads to multiple integrals representations of the correlation functions in the context of which we discuss their long distance asymptotics.

Ken McLaughlin
University of North Carolina

The complete asymptotic expansion of the partition function of Random Matrix Theory: via Riemann-Hilbert techniques

We have obtained a rigorous proof of a complete asymptotic expansion for the partition function of random matrix theory. The original motivation for this expansion was in the physical theory of two dimensional quantum gravity, to assist in enumeration of maps on surfaces with given genus. I will explain the asymptotic expansion, its connection to the combinatorics of maps, and the core ideas of the analysis. This is joint work with Nick Ercolani.

Atsushi Nakayashiki

Classifying the local operators in SU(2) invariant Thirring model

The space of local operators in SU(2) invariant Thirring model (SU(2) ITM) is studied by the form factor bootstrap method. By constructing explicitly sets of form factors we define a subspace of operators which has, by spins of operators, the same character as the level one integrable highest weight representation of the affine Lie algebra of sl(2). This makes a correspondence between the space of local operators in the SU(2) ITM and that in the underlying conformal field theory, the su(2) Wess-Zumino-Witten model at level one.

Andrei Okounkov

Random partitions and their applications


A. Soshnikov
UC Davis, USA

Janossy densities in determinantal and Pfaffian ensembles of random matrices

S-M. Fei
Bonn, Germany

Integrable stochastic lattice systems

K. Takasaki

Landau-Lifshitz equation, elliptic AKNS hierarchy and Sato Grassmannian

C. Klein, J. Frauendiener
MPI, Munchen, Germany and Tubingen, Germany

Hyperelliptic theta functions and spectral methods

E. Zubova, L.I. Manevitch
N.N. Semenov ICP, Moscow, Russia

Alternative approach to the construction of many-solitonic solutions of the sine-gordon equation

I. Calvo, F. Falceto
Zaragoza, Spain

Topological Poisson-sigma models on Poisson-Lie groups

I.V. Krasovsky

Some computable Wiener-Hopf determinants and related polynomials orthogonal on an arc of the unic circle

V. Shramchenko
Concordia, Canada

New Frobenius structures on Hurwitz spaces in terms of Schiffer and Bergmann kernels

B. Bambah
Hyderabad, India

Coherent states of superintegrable systems based on polynomial algebras

V. Golubeva
RIS and TI, Moscow, Russia

On universal R-matrices for two-parametric models with reflections in quantum field theory