A family of random variables X(s), depending on a real parameter s>−12, appears in the asymptotics of the joint moments of characteristic polynomials of random unitary matrices and their derivatives, in the ergodic decomposition of the Hua-Pickrell measures and conjecturally in the asymptotics of the joint moments of Hardy's function and its derivative. Our first main result establishes a connection between the characteristic function of X(s) and the σ-Painlevé III' equation in the full range of parameter values s>−12. Our second main result gives the first explicit expression for the density and all the complex moments of the absolute value of X(s) for integer values of s. Finally, we establish an analogous connection to another special case of the σ-Painlevé III' equation for the Laplace transform of the sum of the inverse points of the Bessel point process.
|Number of pages||30|
|Publication status||Published - 23 Feb 2021|