Exchangeable Arrays and Integrable Systems for Characteristic Polynomials of Random Matrices

ABSTRACT The joint moments of the derivatives of the characteristic polynomial of a random unitary matrix, and also a variant of the characteristic polynomial that is real on the unit circle, in the large matrix size limit, have been studied intensively in the past 25 years, partly in relation to conjectural connections to the Riemann ‐function and Hardy's function. We completely settle the most general version of the problem of convergence of these joint moments, after they are suitably rescaled, for an arbitrary number of derivatives and with arbitrary positive real exponents. Our approach relies on a hidden, higher‐order exchangeable structure, that of an exchangeable array, which, as far as we know, had never been used before in the study of characteristic polynomials of random matrices. We then develop a systematic method, based on a class of Hankel determinants shifted by partitions, that allows us for the first time to give an exact representation of all these joint moments, for finite matrix size, in terms of derivatives of ‐ Painlevé V transcendents. As an application, we can also represent all the joint moments of power sum linear statistics of a certain determinantal point process behind this problem in terms of derivatives of ‐Painlevé III' transcendents. This gives an efficient way to compute all these quantities explicitly. Our methods can be used to obtain analogous results for a number of other models sharing the same features.