Gaussian fluctuations of the determinant of wigner matrices

Paul Bourgade, Krishnan Mody

Research output: Contribution to journalArticle

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 languageEnglish (US)
Article number96
JournalElectronic Journal of Probability
Volume24
DOIs
StatePublished - 2019

Keywords

  • Central limit theorem
  • Determinant
  • Random matrices

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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