论文标题
在一个杰出的随机变量和painlevé方程式上
On a distinguished family of random variables and Painlevé equations
论文作者
论文摘要
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 Hardy功能及其衍生物的共同时刻。我们的第一个主要结果在$ \ mathbf {x}（s）$的特征函数与$σ$-PainlevéIIII'EQUITATION之间的特征函数之间建立了连接。我们的第二个主要结果给出了$ s $的整数值$ \ mathbf {x}（s）$的密度和所有复杂矩的第一个显式表达式。最后，我们建立了与$σ$-PainlevéIII'方程的另一个特殊情况的类似连接，用于贝塞尔点过程的逆点的拉普拉斯变换。
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 $σ$-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 $σ$-Painlevé III' equation for the Laplace transform of the sum of the inverse points of the Bessel point process.
