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].
Highlights
We address the universality of the determinant of a class of random Hermitian matrices
If one were to prove an analogous convergence for the Gaussian Orthogonal Ensemble (GOE), our proof of Theorem 1.2 would extend the result to real symmetric Wigner matrices as well
[44] which describe the fluctuations of individual eigenvalues in the Gaussian Unitary Ensemble (GUE) and GOE
Summary
We address the universality of the determinant of a class of random Hermitian matrices. Before discussing results specific to this symmetry assumption, we give a brief history of results in the non-Hermitian setting. In both settings, a priori bounds preceded estimates on moments of determinants, and the distribution of determinants for integrable models of random matrices. A priori bounds preceded estimates on moments of determinants, and the distribution of determinants for integrable models of random matrices The universality of such determinants has been the subject of recent active research
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