Research in Random Matrices and Integrable Systems

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PI: Harold Widom, UC Santa CruzDMS-0243982*********************************************************************There are four projects, the first three joint with Craig A. Tracy and the fourth joint with Estelle L. Basor. The first is to search for a system of partial differential equations for the finite-dimensional distribution functions for the Airy process, which is expected to describe all growth processes in the KPZ universality class. (This has already been accomplished.) The second project is to find limit theorems for the Hall-Littlewood polynomials, interpolating known results for the Schur functions (Johansson) and the shifted Schur functions (Tracy and the PI). New methods are needed here because of a lack of determinantal representation. The third project is to complete earlier work on the asymptotics of solutions to the periodic Toda equations by determining the asymptotics on the critical curves, where the asymptotics will take a very different form. The fourth project is to prove correct the conjectured constant in an asymptotic expansion of Dyson related to the spacings between eigenvalues of random matrices in the GUE universality class. To this end it is hoped to refine the method recently used by Basor and the PI to obtain the asymptotics of Wiener-Hopf determinants with Fisher-Hartwig symbols. Many physical systems possess such complicated behavior that exact predictions become impossible. Random matrix theory provides mathematical models that allow a simulation of such behavior and predictions that allow comparison with experiment. This was its original motivation, but it has since had significant applications in other areas of mathematics, science and technology. Work of the PI (largely collaborative) in this subject has had application in statistics where it has been applied to the analysis of large data sets, in communications where it has been used to study the start-up behavior of a long production line or the transient flow of messages over a long path in a communications network, and in physics and technology where it has had an impact on the theory of gap fluctuations in a metal grain or quantum dot. This project is concerned with different aspects of the theory including the search for the solution to unsolved problems. The project is also concerned with the connection of random matrix theory withcombinatorics and growth processes. This is a more recent development that could shed light on the question of universality of probability distributions related to the Airy process mentioned above.