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Annales Henri Poincaré

, Volume 20, Issue 1, pp 259–298 | Cite as

Mod-Gaussian Convergence for Random Determinants

  • Martina Dal BorgoEmail author
  • Emma Hovhannisyan
  • Alain Rouault
Article
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Abstract

The aim of this paper is to give a precise asymptotic description of some eigenvalue statistics stemming from random matrix theory. More precisely, we consider random determinants of the GUE, GOE, Laguerre, uniform Gram and Jacobi beta ensembles and random characteristic polynomials evaluated at 1 for matrices in the circular and circular Jacobi beta ensembles. We use the framework of mod-Gaussian convergence to provide quantitative estimates of their logarithmic behavior, as the size of the ensemble grows to infinity. We establish central limit theorems, Berry–Esseen bounds, moderate deviations and local limit theorems. Furthermore, we identify the scale at which the validity of the Gaussian approximation for the tails breaks. With the exception of the Gaussian ensemble, all the results are obtained for a continuous choice of the Dyson parameter, that is, for general \(\beta >0\). The proofs rely on explicit computations which are possible thanks to closed product formulas of the Laplace transforms.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Martina Dal Borgo
    • 1
    Email author
  • Emma Hovhannisyan
    • 1
  • Alain Rouault
    • 2
  1. 1.University of ZurichZürichSwitzerland
  2. 2.Laboratoire de Mathématiques de Versailles UVSQ, CNRSUniversité Paris-SaclayVersailles CedexFrance

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