On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices
On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices
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DOI：
10.1007/s00220-019-03503-7
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发表时间：
2019-10-01
影响因子：
2.4
通讯作者：
Keating, J. P.
中科院分区：
文献类型：
--
作者：
Bailey, E. C.;Keating, J. P.
用P-N(A, θ)=det（I-Ae(-i θ)）表示NxN随机酉矩阵A复平面上单位圆上的特征多项式，我们计算随机变量对应于P-N（A, θ）的第k个矩，定义为A上的平均值是U(N)的一个元素，对应于P-N（A, θ）的第2个矩相对于均匀测度d /2，对于所有k， β是N的一个元素。这些矩的矩在最近的特征多项式的极值统计及其与对数相关高斯场的联系的研究中发挥了重要作用。我们的方法是基于使用对称函数理论的力矩的新的组合表示，以及根据多个轮廓积分的第二种表示的分析。我们的主要结果是，矩的矩是N阶k(2) -k+1的多项式。这解决了Fyodorov和Keating （Philos Trans R Soc a 372(2007): 20120503,2014）关于N为N ->无穷大的矩的缩放的猜想，对于k， β是N的一个元素。实际上，它更进一步，我们给出了一种显式计算这些多项式的方法，并获得了主要系数的一般公式。
Denoting by P-N(A,theta)=det(I-Ae(-i theta)) the characteristic polynomial on the unit circle in the complex plane of an NxN random unitary matrix A, we calculate the kth moment, defined with respect to an average over A is an element of U(N), of the random variable corresponding to the 2 beta th moment of P-N(A,theta) with respect to the uniform measure d theta/2 pi, for all k,beta is an element of N . These moments of moments have played an important role in recent investigations of the extreme value statistics of characteristic polynomials and their connections with log-correlated Gaussian fields. Our approach is based on a new combinatorial representation of the moments using the theory of symmetric functions, and an analysis of a second representation in terms of multiple contour integrals. Our main result is that the moments of moments are polynomials in N of degree k(2)beta(2)-k+1. This resolves a conjecture of Fyodorov and Keating (Philos Trans R Soc A 372(2007):20120503, 2014) concerning the scaling of the moments with N as N ->infinity, for k,beta is an element of N. Indeed, it goes further in that we give a method for computing these polynomials explicitly and obtain a general formula for the leading coefficient.