标题： 随机矩阵理论中的对偶性 显示英文标题

标题： Dualities in random matrix theory

摘要： 随机矩阵理论中关于特征多项式的乘积和幂以及矩的对偶恒等式被综述。 一个典型的对偶恒等式结构是，对于特定的$N \times N$矩阵系综的特征多项式的正整数次幂$k$的平均值，它被表示为另一种随机矩阵系综的特征多项式的幂$N$的平均值，现在该系综的大小为$k \times k$。 除了少数例外，这些对偶性涉及（$\beta$一般的）经典高斯、拉盖尔和雅可比厄米特系综，圆雅可比系综，或者与吉尼布雷随机矩阵相关的各种非厄米特系综。 在厄米特情况下具有单位对称性的情况下，可以利用行列式结构进行研究。 $\beta$一般的案例需要使用杰克多项式理论，特别是基于杰克多项式的超几何函数。 还综述了其在计算各种$\beta$系综相关性和分布函数的标度极限中的应用。 非厄米特情况依赖于对应于区域多项式的杰克多项式的特殊情况，以及当它们的参数是某些矩阵的特征值时的积分性质。 研究谱密度矩及其推广的对偶性的主要工具是环方程。 显示英文摘要

摘要： Duality identities in random matrix theory for products and powers of characteristic polynomials, and for moments, are reviewed. The structure of a typical duality identity for the average of a positive integer power $k$ of the characteristic polynomial for particular ensemble of $N \times N$ matrices is that it is expressed as the average of the power $N$ of the characteristic polynomial of some other ensemble of random matrices, now of size $k \times k$. With only a few exceptions, such dualities involve (the $\beta$ generalised) classical Gaussian, Laguerre and Jacobi ensembles Hermitian ensembles, the circular Jacobi ensemble, or the various non-Hermitian ensembles relating to Ginibre random matrices. In the case of unitary symmetry in the Hermitian case, they can be studied using the determinantal structure. The $\beta$ generalised case requires the use of Jack polynomial theory, and in particular Jack polynomial based hypergeometric functions. Applications to the computation of the scaling limit of various $\beta$ ensemble correlation and distribution functions are also reviewed. The non-Hermitian case relies on the particular cases of Jack polynomials corresponding to zonal polynomials, and their integration properties when their arguments are eigenvalues of certain matrices. The main tool to study dualities for moments of the spectral density, and generalisations, is the loop equations.