Density of eigenvalues and its perturbation invariance in unitary ensembles of random matrices

2008/10/30 by Dang-Zheng Liu, Zhengdong Wang, Liu, Dang-Zheng +3
Mathematics · #15A52 #30E05 #60F99 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.0810.5425

openalex publication_date 2008/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generally study the density of eigenvalues in unitary ensembles of random matrices from the recurrence coefficients with regularly varying conditions for the orthogonal polynomials. First we calculate directly the moments of the density. Then, by studying some deformation of the moments, we get a family of differential equations of first order which the densities satisfy (see Theorem 1.2), and give the densities by solving them. Further, we prove that the density is invariant after the polynomial perturbation of the weight function (see Theorem 1.5).

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