标题： 特征多项式相关性的普遍结果：黎曼-希尔伯特方法 显示英文标题

标题： Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

摘要： 我们证明了厄米特随机矩阵的特征多项式的比值和乘积的一般关联函数由三种不同类型的可积核所支配：a) 由正交多项式构造的核；b) 由相同正交多项式的柯西变换构造的核；最后是c) 由正交多项式及其柯西变换共同构造的核。 这些核与正交多项式的黎曼-希尔伯特问题相关。 对于关联函数，我们得到了这些核行列式的精确表达式。 推导出的表示使我们能够通过黎曼-希尔伯特问题的Deift-Zhou最陡下降/驻相法研究特征多项式的关联函数渐近行为，并且特别地找到特征多项式的负矩。 这揭示了任意具有$\beta=2$对称类的不变系综中特征多项式的关联函数和矩的普遍部分。 显示英文摘要

摘要： We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of $\beta=2$ symmetry class.