Random Matrices and Integrability [Batsheva Fellows: Abstracts]

Lectures sponsored by The Batsheva de Rothschild Fund

The Israel Academy of Sciences and Humanities has nominated Alexander R. Its, Distinguished Professor at Indiana University-Purdue University Indianapolis, and Craig A. Tracy, Distinguished Professor at University of California (Davis), for the Batsheva Fellowship in Natural Sciences and Mathematics.

The asymmetric simple exclusion process: Integrable structure and limit theorems / Craig A. Tracy @ Technion

Since its introduction nearly forty years ago, the asymmetric simple exclusion process (ASEP), has become the "default stochastic model for transport phenomena". Some have called the ASEP the "Ising model for nonequilibrium physics". In ASEP on the integer lattice particles move according to two rules: (1) A particle at waits an exponential time with parameter one (independently of all the other particles), and then it chooses with probability ; (2) If is vacant at that time it moves to , while if is occupied it remains at and restarts the clock. The adjective "simple" refers to the fact that allowed jumps are one step to the right, , or one step to the left, . The asymmetric condition means so that there is a net drift to either the right or the left.

In this lecture we consider ASEP on the integer lattice with step initial condition: At time zero the particles are located at and there is a drift to the left (). If denotes the position of the th particle from the left at time (so that ), a basic quantity is the distribution function which describes the "current fluctuations". Physicists have conjectured that the limiting distribution of as with fixed is in the Kardar-Parisi-Zhang Universality Class. We show that this is indeed the case and describe the limiting distribution function. (This limiting distribution function first appeared in the random matrix theory literature.) This result extends an earlier theorem of Kurt Johansson on the T(totally)ASEP where and . This work is joint work with Harold Widom. The lecture itself is for a general audience.

The Riemann-Hilbert method / Alexander Its @ Tel Aviv University

In this talk a general overview of the Riemann-Hilbert method, which was originated in 1970s−1980s in the theory of integrable nonlinear partial differential equations of the KdV type, will be given. The most recent applications of the Riemann-Hilbert approach to asymptotic problems arising in the theory of matrix models, combinatorics and integrable statistical mechanics will be outlined.

The asymmetric simple exclusion process: Integrable structure and limit theorems / Craig A. Tracy @ Hebrew University

Since its introduction nearly forty years ago, the asymmetric simple exclusion process (ASEP), has become the "default stochastic model for transport phenomena". Some have called the ASEP the "Ising model for nonequilibrium physics". In ASEP on the integer lattice particles move according to two rules: (1) A particle at waits an exponential time with parameter one (independently of all the other particles), and then it chooses with probability ; (2) If is vacant at that time it moves to , while if is occupied it remains at and restarts the clock. The adjective "simple" refers to the fact that allowed jumps are one step to the right, , or one step to the left, . The asymmetric condition means so that there is a net drift to either the right or the left.

In this lecture we consider ASEP on the integer lattice with step initial condition: At time zero the particles are located at and there is a drift to the left (). If denotes the position of the th particle from the left at time (so that ), a basic quantity is the distribution function which describes the "current fluctuations". Physicists have conjectured that the limiting distribution of as with fixed is in the Kardar-Parisi-Zhang Universality Class. We show that this is indeed the case and describe the limiting distribution function. (This limiting distribution function first appeared in the random matrix theory literature.) This result extends an earlier theorem of Kurt Johansson on the T(totally)ASEP where and . This work is joint work with Harold Widom. The lecture itself is for a general audience.

Special functions and integrable systems / Alexander Its @ Hebrew University

The recent developments in the theory of integrable systems have revealed its intrinsic relation to the theory of special functions. Perhaps the most generally known aspects of this relation are the group-theoretical, especially the quantum-group theoretical, and the algebra-geometrical ones. In the talk we will discuss the analytic side of the "special functions−integrable systems" connection. This aspect of the relation between the two theories is less known to the general mathematical community, although it goes back to the classical works of Fuchs, Garnier and Schlesinger on the isomonodrony deformations of the systems of linear differential equations with rational coefficients. Indeed, the monodromy theory of linear systems provides a unified framework for the linear (hypergeometric type) and nonlinear (Painlevé type) special functions and, simultaneously, builds a base for the new powerful technique of the asymptotic analysis − the Riemann-Hilbert method.

In this survey talk, which is based on the works of many authors spanned over more than two decades, the isomonodromy point of view on special function will be outlined. We will also review the history of the Riemann-Hilbert method as well as its most recent applications in the theory of orthogonal polynomials and random matrices.

Integrable models in statistical physics and associated universality theorems and conjectures / Craig A. Tracy @ Holon Institute of Technology

This lecture, designed for a general audience, will survey "exactly solvable" models in statistical physics. The three main examples will be the 2D Ising model, Random Matrix Models, and the Asymmetric Simple Exclusion Process. The underlying theme is the connection with integrable differential equations of Painlevé type.

Painlevé transcendents and random matrices / Alexander Its @ Weizmann Institute

Painlevé functions have been playing increasingly important role in random matrix theory since the early nineties works on the 2d quantum gravity of Brézin and Kazakov, Douglas and Shenker, and Gross and Migdal, and the works (also early nineties) of Mehta and Mahoux, and Tracy and Widom on the level spacing distributions. Indeed, the appearance of Painlevé functions in the field can be traced to 1970s−1980s works of Barouch, McCoy, Tracy, and Wu, and of Jimbo, Miwa, Môri and Sato on the quantum correlation functions. In this talk we will try to review these, and also some of the more recent results concerning Painlevé transcendents and random matrices. We will present a unified point view on the subject based on the Riemann-Hilbert method. The emphasis will be made on the various double scaling limits related to the universal properties of random matrices for which Painlevé functions provide an adequate "special function environment".