On a distinguished family of random variables and Painlevé equations

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 $\sigma$-Painlevé 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 $\sigma$-Painlevé III' equation for the Laplace transform of the sum of the inverse points of the Bessel point process.

arXiv commentImprovements in exposition and a number of references added. To appear PMP

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Cited by in corpus (1)

• Convergence and an explicit formula for the joint moments of the Circular Jacobi $\beta$-Ensemble characteristic polynomial