Asymptotic Independence of the Extreme Eigenvalues of Gue

We give a short, operator-theoretic proof of the asymptotic independence 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 those extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. The method is amenable to explicitly establish the leading order term of an asymptotic expansion. As a corollary we obtain that the correlation of the extreme eigenvalues asymptotically behaves like n−2/3/4σ2, where σ2 denotes the variance of the Tracy–Widom distribution. We consider the n× n Gaussian Unitary Ensemble (GUE) with the joint probability distribution of its (unordered) eigenvalues given by pn(λ1, . . . , λn) = cne 2 1−···−λ 2 n ∏ i<j |λi − λj| and denote the induced random variables of the minimal and maximal eigenvalue by λ (n) min and λ (n) max. Bianchi, Debbah and Najim (2008) have recently shown the asymptotic independence of the edge-scaled extreme eigenvalues, that is, P ( λ̃ (n) min 6 x, λ̃ (n) max 6 y ) = P ( λ̃ (n) min 6 x ) ·P ( λ̃ (n) max 6 y ) + o(1) (n→ ∞) (1)