On a distinguished family of random variables and Painlev\'e equations

TL;DR

This paper explores a family of random variables linked to Painlevé equations, providing explicit formulas for their densities and moments, and connecting them to random matrix theory and special functions.

Contribution

It establishes a novel connection between the characteristic function of these variables and the Painlevé III' equation across all relevant parameters, and derives explicit density and moments for integer parameters.

Findings

Connection between characteristic function and Painlevé III' equation

Explicit density and moments for integer s

Link to Bessel point process inverse points

Abstract

A family of random variables $\mathbf{X}(s)$, depending on a real parameter $s>-\frac{1}{2}$, 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 $\mathbf{X}(s)$ and the $\sigma$-Painlev\'e III' equation in the full range of parameter values $s>-\frac{1}{2}$. Our second main result gives the first explicit expression for the density and all the complex moments of the absolute value of $\mathbf{X}(s)$ for integer values of $s$. Finally, we establish an analogous connection to another special case of the $\sigma$-Painlev\'e III' equation for the Laplace transform of the sum of the inverse points of the Bessel point process.