# Asymptotic Independence of the Extreme Eigenvalues of GUE - Mathematics > Probability

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Abstract: We give a short, operator-theoretic proof of the asymptotic independenceincluding a first correction term of the minimal and maximal eigenvalue ofthe n \times n Gaussian Unitary Ensemble in the large matrix limit n \to\infty. This is done by representing the joint probability distribution of theextreme eigenvalues as the Fredholm determinant of an operator matrix thatasymptotically becomes diagonal. As a corollary we obtain that the correlationof the extreme eigenvalues asymptotically behaves like n^{-2-3}-4\sigma^2,where \sigma^2 denotes the variance of the Tracy-Widom distribution. While weconjecture that the extreme eigenvalues are asymptotically independent forWigner random hermitian matrix ensembles in general, the actual constant in theasymptotic behavior of the correlation turns out to be specific and can thus beused to distinguish the Gaussian Unitary Ensemble statistically from otherWigner ensembles.

Author: ** Folkmar Bornemann**

Source: https://arxiv.org/