Asymptotic Independence of the Extreme Eigenvalues of GUE

Abstract

We give a short, operator-theoretic proof of the asymptotic independence (including a first correction term) of the minimal and maximal eigenvalue of the n \times n Gaussian Unitary Ensemble in the large matrix limit n \to \infty. This is done by representing the joint probability distribution of the extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. As a corollary we obtain that the correlation of the extreme eigenvalues asymptotically behaves like n^{-2/3}/4σ^2, where σ^2 denotes the variance of the Tracy--Widom distribution. While we conjecture that the extreme eigenvalues are asymptotically independent for Wigner random hermitian matrix ensembles in general, the actual constant in the asymptotic behavior of the correlation turns out to be specific and can thus be used to distinguish the Gaussian Unitary Ensemble statistically from other Wigner ensembles.

added comments on universality and a numerical experiment with uniformely distributed matrix entries, 7 pages

References in corpus (3)

• On the Numerical Evaluation of Fredholm Determinants
• On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
• Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited

Cited by in corpus (3)

• Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges
• Limiting distribution of maximal crossing and nesting of Poissonized random matchings
• Asymptotic expansions relating to the distribution of the length of longest increasing subsequences