Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach
Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach
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特征多项式相关性的通用结果：黎曼-希尔伯特方法
DOI：
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发表时间：
2002
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通讯作者：
Y. Fyodorov
中科院分区：
文献类型：
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作者：
E. Strahov;Y. Fyodorov
本文证明了Hermitian随机矩阵的特征多项式的比和积的一般相关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可积核，B）由同一正交多项式的Cauchy变换构造的可积核，以及c）由正交多项式及其Cauchy变换构造的可积核.这些核与正交多项式的Riemann-Hilbert问题有关。对于相关函数，我们得到这些内核的行列式的形式的精确表达式。导出表示使我们能够通过Deift-Zhou最速下降/驻相方法研究Riemann-Hilbert问题的特征多项式的相关函数的渐近性，特别是找到特征多项式的负矩。这揭示了β=2对称类的任意不变系综的相关函数和特征多项式矩的普适部分。
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) those 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 the 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 an arbitrary invariant ensemble of β=2 symmetry class.