论文标题
正交和符号随机矩阵的特征多项式，Jacobi合奏和L函数
Characteristic Polynomials of Orthogonal and Symplectic Random Matrices, Jacobi Ensembles & L-functions
论文作者
论文摘要
从蒙哥马利的猜想开始，对随机矩阵理论和L功能理论的联系一直引起人们的重大兴趣。特别是，在各种著作中考虑了随机矩阵的特征多项式矩，以估计L功能家族的矩形渐近学。在本文中，我们首先考虑了符号随机矩阵及其第二个衍生物的特征多项式的关节力矩。我们获得了渐近级系数的渐近系数，就painleve方程的溶液表示。这为我们提供了关于Dirichlet L功能家族的相应关节力矩的猜想渐近学。在这样做的过程中，我们计算了某种添加剂雅各比统计量的渐近学，这可能是随机矩阵理论的独立兴趣。最后，我们考虑了一个略有不同类型的关节力矩，这是以前在各种作品中考虑的平均值的平均值的类似物。我们明确地获得渐近系数和领先顺序系数。
Starting from Montgomery's conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been considered in various works to estimate the asymptotics of moments of L-function families. In this paper, we first consider joint moments of the characteristic polynomial of a symplectic random matrix and its second derivative. We obtain the asymptotics, along with a representation of the leading order coefficient in terms of the solution of a Painleve equation. This gives us the conjectural asymptotics of the corresponding joint moments over families of Dirichlet L-functions. In doing so, we compute the asymptotics of a certain additive Jacobi statistic, which could be of independent interest in random matrix theory. Finally, we consider a slightly different type of joint moment that is the analogue of an average considered over the unitary group in various works before. We obtain the asymptotics and the leading order coefficient explicitly.
