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2019 Gaussian fluctuations of the determinant of Wigner matrices
Paul Bourgade, Krishnan Mody
Electron. J. Probab. 24: 1-28 (2019). DOI: 10.1214/19-EJP356


We prove that the logarithm of the determinant of a Wigner matrix satisfies a central limit theorem in the limit of large dimension. Previous results about fluctuations of such determinants required that the first four moments of the matrix entries match those of a Gaussian [53]. Our work treats symmetric and Hermitian matrices with centered entries having the same variance and subgaussian tail. In particular, it applies to symmetric Bernoulli matrices and answers an open problem raised in [54]. The method relies on (1) the observable introduced in [9] and the stochastic advection equation it satisfies, (2) strong estimates on the Green function as in [12], (3) fixed energy universality [10], (4) a moment matching argument [52] using Green’s function comparison [21].


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Paul Bourgade. Krishnan Mody. "Gaussian fluctuations of the determinant of Wigner matrices." Electron. J. Probab. 24 1 - 28, 2019.


Received: 29 May 2019; Accepted: 26 August 2019; Published: 2019
First available in Project Euclid: 18 September 2019

zbMATH: 07107403
MathSciNet: MR4017114
Digital Object Identifier: 10.1214/19-EJP356

Primary: 15B52


Vol.24 • 2019
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