论文标题
对称函数理论和统一不变合奏
Symmetric Function Theory and Unitary Invariant Ensembles
论文作者
论文摘要
代表理论和对称函数的理论在计算数量的计算中，在随机矩阵理论中起着核心作用，例如痕迹的关节矩以及从圆形单一集合和其他与经典紧凑型组相关的圆形单一集合和其他圆形合奏中绘制的特征多项式的关节矩。原因是它们可以启用精确公式的推导，然后提供了计算这些数量的大矩阵渐近性的途径。我们为随机矩阵的高斯单位合奏和其他相关的单一不变矩阵合奏开发了平行理论。这使我们可以在这些情况下写下精确的公式，以根据适当定义的对称函数的痕迹的关节力矩和特征多项式的关节力矩。作为应用的一个例子，对于痕迹的关节力矩，我们为矩阵大小趋向于无限属时，为GUE矩阵的多项式函数的收敛速率提供了明确的渐近公式。
Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the Circular Unitary Ensemble and other Circular Ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulae, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble of random matrices, and other related unitary invariant matrix ensembles. This allows us to write down exact formulae in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces we derive explicit asymptotic formulae for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.
