Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory

作者：Pan, Guangming; Wang, Shaochen^*; Zhou, Wang

来源：Journal of Mathematical Physics, 2017, 58(10): 103301.

DOI：10.1063/1.5006507

• LOCAL EIGENVALUE STATISTICS
• WIGNER MATRICES
• GAUSSIAN FLUCTUATIONS
• MODERATE DEVIATIONS
• UNIVERSALITY
• BOUNDS
• EDGE
• CLT

In this paper, we consider the asymptotic behavior of X-fn((n)) := Sigma(n)(i=1) f(n)(x(i)), where x(i), i = 1, ... , n form orthogonal polynomial ensembles and f(n) is a real-valued, bounded measurable function. Under the condition that VarX(fn)((n)) -> infinity, the Berry-Esseen (BE) bound and Cram ' er type moderate deviation principle (MDP) for X-fn((n)) are obtained by using the method of cumulants. As two applications, we establish the BE bound and Cramer type MDP for linear spectrum statistics of Wigner matrix and sample covariance matrix in the complex cases. These results show that in the edge case (which means f(n) has a particular form f(x)I(x >= theta(n)) where theta(n) is close to the right edge of equilibrium measure and f is a smooth function), X-fn((n)) behaves like the eigenvalues counting function of the corresponding Wigner matrix and sample covariance matrix, respectively. Published by AIP Publishing.

• 出版日期2017-10
• 单位华南理工大学