Consider a $n times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(Delta_{i,n}, 1leq ileq p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wign
ers semi-circle law, we prove that the related counting measures $({mathcal N}_n(Delta_{i,n}), 1leq ileq p)$, where ${mathcal N}_n(Delta)$ represents the number of eigenvalues within $Delta$, are asymptotically independent as the size $n$ goes to infinity, $p$ being fixed. As a consequence, we prove that the largest and smallest eigenvalues, properly centered and rescaled, are asymptotically independent; we finally describe the fluctuations of the condition number of a matrix from the GUE.
In this paper, using tools from asymptotic random matrix theory, a new cooperative scheme for frequency band sensing is introduced for both AWGN and fading channels. Unlike previous works in the field, the new scheme does not require the knowledge of
the noise statistics or its variance and is related to the behavior of the largest and smallest eigenvalue of random matrices. Remarkably, simulations show that the asymptotic claims hold even for a small number of observations (which makes it convenient for time-varying topologies), outperforming classical energy detection techniques.
