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引用次数: 0
摘要
从蒙哥马利猜想开始,人们对随机矩阵理论和 L 函数理论之间的联系产生了浓厚的兴趣。特别是,随机矩阵的特征多项式的矩在各种著作中被用来估计 L 函数族的矩的渐近性。在本文中,我们首先考虑交点随机矩阵的特征多项式及其二次导数的联合矩。我们得到了渐近线,以及前阶系数在潘列韦方程解中的表示。这样,我们就得到了迪里夏特 L 函数族上相应联合矩的猜想渐近学。在此过程中,我们计算了某个加性雅可比统计量的渐近线,这可能与随机矩阵理论有关。最后,我们考虑了一种略有不同的联合矩,它是之前各种著作中考虑的 U(N) 上平均值的类似物。我们明确地得到了渐近线和前阶系数。
Characteristic polynomials of orthogonal and symplectic random matrices, Jacobi ensembles & L-functions
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 Painlevé 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 in various works before. We obtain the asymptotics and the leading order coefficient explicitly.
期刊介绍:
Random Matrix Theory (RMT) has a long and rich history and has, especially in recent years, shown to have important applications in many diverse areas of mathematics, science, and engineering. The scope of RMT and its applications include the areas of classical analysis, probability theory, statistical analysis of big data, as well as connections to graph theory, number theory, representation theory, and many areas of mathematical physics.
Applications of Random Matrix Theory continue to present themselves and new applications are welcome in this journal. Some examples are orthogonal polynomial theory, free probability, integrable systems, growth models, wireless communications, signal processing, numerical computing, complex networks, economics, statistical mechanics, and quantum theory.
Special issues devoted to single topic of current interest will also be considered and published in this journal.