Australian Mathematical Society
Mathematics Department at Macquarie University

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50th Annual Meeting of the Australian Mathematical Society

Plenary Talk in Macquarie Theatre

Monday 25 September 2006 at 11:30


Peter Forrester (University of Melbourne)


Random Matrices and Painlevé Transcendents


Random matrix theory has applications in a number of diverse fields. For example, in analytic number theory the Montgomery-Odlyzko law states that the statistics of the large zeros of the Riemann zeta function on the critical line coincide with the statistics of the eigenvalues of large random unitary matrices. In combinatorics the distribution of the longest increasing subsequence of a large random permutation coincides with the distribution of the largest eigenvalue of a large random complex Hermitian matrix. Quantitative results for eigenvalue distributions, and thus for the distributions in the applications, can be obtained in terms of Painlevé transcendents. We will emphasize the approach to such calculations via a Hamiltonian formulation of Painlevé systems.