Applied Mathematics Seminar

Having originated in mathematical statistics and nuclear physics, random matrix theory now occupies a unique position at the crossroads of many theoretical systems in mathematics and physics. It is the theoretical paradigm for quantum systems whose underlying classical dynamics is strongly chaotic and explains the universal features found in them (and forms the basis of an unproven conjecture by Bohigas, Gianonni and Schmit). Other applications in physics include 2-D quantum gravity, quantum chromodynamics, transport in disordered mesoscopic systems and integrable hierarchies such as the Kadomtsev-Petviashvili hierarchy to name a few.

In mathematics there are numerous combinatorial problems---shape fluctuations in random growth models, random tilings, longest increasing subsequences in random words, random permutations and involutions, vicious random walker models, random Young tableaux, etc.---that are equivalent to a random matrix problem. And it is known to great accuracy that the local spacing distribution of the high zeros of the Riemann zeta function on the critical line follow those of the eigenvalues of large random hermitian matrices (the Montgomery-Odlyzko conjecture), and this constitutes the most convincing evidence for a spectral interpretation of these zeros.

However, in this talk I wish to discuss another issue---the connection with integrable dynamical systems, which has been the subject of recent work done in collaboration with Peter Forrester (Melbourne) and Chris Cosgrove (Sydney). There is an intimate connection between classes of random matrix ensembles associated with the classical orthogonal polynomial systems, and nonlinear integrable dynamical systems of the Painlev\'e type. This correspondence will be illustrated with the example of the Gaussian unitary ensemble and the Painlev\'e transcendents P-IV and P-II utilising the Okamoto and Noumi-Yamada formulation of Painlev\'e theory, although it is known to be generally true that the probability of finding a gap in the spectrum of matrix ensembles is a $\tau$-function in the Painlev\'e theory. Some new perspectives on Painlev\'e theory will be presented and a number of open questions will be posed as a consequence of this.