University of Birmingham > Talks@bham > Mathematics Colloquium > Characteristic Polynomials of Random Unitary Matrices, Partition Sums, and Painlevé Equations

Characteristic Polynomials of Random Unitary Matrices, Partition Sums, and Painlevé Equations

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Tea will be served in the common room before the talk.

The moments of characteristic polynomials play a central role in Random Matrix Theory. They appear in many applications, ranging from quantum mechanics to number theory. The mixed moments of the characteristic polynomials of random unitary matrices, i.e. the joint moments of the polynomials and their derivatives, can be expressed recursively in terms of combinatorial sums involving partitions. These combinatorial sums are not easy to compute, however, and so this does not give an effective method for calculating the mixed moments in general. I shall describe a new, alternative evaluation, in terms of solutions of Painlevé differential equations, that facilitates their computation and which allows one to prove previous conjectures concerning their asymptotics when the matrices are large.

This talk is part of the Mathematics Colloquium series.

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