Asymptotic independence of the extreme eigenvalues of Gaussian unitary ensemble.

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×n Gaussian unitary ensemble in the large matrix limit n→∞. 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 get that the correlation of the extreme eigenvalues asymptotically behaves like n-2/3/4σ2, where σ2 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 certain other Wigner ensembles. [ABSTRACT FROM AUTHOR]