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 . 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 . The method relies on (1) the observable introduced in  and the stochastic advection equation it satisfies, (2) strong estimates on the Green function as in , (3) fixed energy universality , (4) a moment matching argument  using Green’s function comparison .
- Central limit theorem
- Random matrices
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty