A central limit theorem for the determinant of a Wigner matrix

2011/11/27 by Terence Tao, Van Vu, Tao, Terence +1
Mathematics · #15A52 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1111.6300

openalex publication_date 2011/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a central limit theorem for the log-determinant log|det(M_n)| of a Wigner matrix M_n, under the assumption of four matching moments with either the GUE or GOE ensemble. More specifically, we show that this log-determinant is asymptotically distributed like N(log √(n!) - 1/2 log n, 1/2 log n)_\R when one matches moments with GUE, and N(log √(n!) - 1/4 log n, 1/4 log n)_\R when one matches moments with GOE.

Citations

• The Distribution of the Determinant of a Complex Wishart Distributed Matrix

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• An edge CLT for the log determinant of Wigner ensembles
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• A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices
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