### Abstract

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].

Original language | English (US) |
---|---|

Article number | 96 |

Journal | Electronic Journal of Probability |

Volume | 24 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Central limit theorem
- Determinant
- Random matrices

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Electronic Journal of Probability*,

*24*, [96]. https://doi.org/10.1214/19-EJP356

**Gaussian fluctuations of the determinant of wigner matrices.** / Bourgade, Paul; Mody, Krishnan.

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 24, 96. https://doi.org/10.1214/19-EJP356

}

TY - JOUR

T1 - Gaussian fluctuations of the determinant of wigner matrices

AU - Bourgade, Paul

AU - Mody, Krishnan

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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].

AB - 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].

KW - Central limit theorem

KW - Determinant

KW - Random matrices

UR - http://www.scopus.com/inward/record.url?scp=85073276166&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85073276166&partnerID=8YFLogxK

U2 - 10.1214/19-EJP356

DO - 10.1214/19-EJP356

M3 - Article

AN - SCOPUS:85073276166

VL - 24

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 96

ER -