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

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. However, these combinatorial sums are not easy to compute, and so this does not give an effective method for calculating the mixed moments in general. I shall describe an alternative evaluation of the mixed moments, in terms of solutions of the Painlevé V differential equation, that facilitates their computation and asymptotic analysis.