Extreme value statistics of eigenvalues of Gaussian random matrices

David S. Dean and Satya N. Majumdar
Phys. Rev. E 77, 041108 – Published 10 April 2008

Abstract

We compute exact asymptotic results for the probability of the occurrence of large deviations of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N×N) random matrix are positive (negative) decreases for large N as exp[βθ(0)N2] where the Dyson index β characterizes the ensemble and the exponent θ(0)=(ln3)/4=0.274653 is universal. We compute the probability that the eigenvalues lie in the interval [ζ1,ζ2] which allows us to calculate the joint probability distribution of the minimum and the maximum eigenvalue. As a by-product, we also obtain exactly the average density of states in Gaussian ensembles whose eigenvalues are restricted to lie in the interval [ζ1,ζ2], thus generalizing the celebrated Wigner semi-circle law to these restricted ensembles. It is found that the density of states generically exhibits an inverse square-root singularity at the location of the barriers. These results are confirmed by numerical simulations. Some of the results presented in detail here were announced in a previous paper [D. S. Dean and S. N. Majumdar, Phys. Rev. Lett. 97, 160201 (2006)].

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Received 10 January 2008

DOI:https://doi.org/10.1103/PhysRevE.77.041108

©2008 American Physical Society

Authors & Affiliations

David S. Dean1 and Satya N. Majumdar2

  • 1Laboratoire de Physique Théorique (UMR 5152 du CNRS), Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex 4, France
  • 2Laboratoire de Physique Théorique et Modèles Statistiques (UMR 8626 du CNRS), Université Paris-Sud, Bât. 100, 91405 Orsay Cedex, France

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 77, Iss. 4 — April 2008

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review E

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×