Point Processes, Noise, and Stochastic Analysis (lectures by Balint Vir´ag )

Let M be an n × n matrix with i.i.d. CN(0, 1) entries (Complex Normal random variables with mean 0 and variance 1), and define A = M +M ∗ √ 2 , where M∗ denotes the conjugate transpose of M . The resulting random matrix is a member of the Gaussian Unitary Ensemble (GUE), and we would like to classify the distribution of its eigenvalues. In a previous section, the measure of the Gaussian Unitary Ensemble was classified as a certain product measure. Namely, given a real function f on n× n matrices and A as above, we have E[f(A)] = ∫ f(ODΛO)|∆|e ∑n i=1 λ 2 i (Π dλi) dO, where dλi is Lebesgue measure on R, dO is Haar measure on the Orthogonal Group of degree n, DΛ is the diagonal matrix with entries λ1, . . . , λn, and |∆| is the determinant of the so called Vandermonde Matrix, ∆i,j = λ i−1 j , which was shown to be |∆| = ∏