2021/12/21 by László Erdős, Erdős, László, Hong Chang Ji +1
Mathematics · Physics and Astronomy · #15B52 #60B20 #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Random Matrices and Applications #math.PR #msc:15B52 #msc:60B20
paper · pdf · doi:10.48550/arxiv.2112.11382
23 pages
arxiv created 2021/12/21 · openalex publication_date 2021/12/21 · arxiv updated 2021/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For large dimensional non-Hermitian random matrices X with real or complex independent, identically distributed, centered entries, we consider the fluctuations of f(X) as a matrix where f is an analytic function around the spectrum of X. We prove that for a generic bounded square matrix A, the quantity Trf(X)A exhibits Gaussian fluctuations as the matrix size grows to infinity, which consists of two independent modes corresponding to the tracial and traceless parts of A. We find a new formula for the variance of the traceless part that involves the Frobenius norm of A and the L2-norm of f on the boundary of the limiting spectrum.