}∞ i,j=1 be i.i.d. standard Gaussian random variables. Definition The Gaussian Unitary Ensemble (GUE) is the space of all n × n matrices of the form: Xn = x1,1 x1,2+iy1,2 √ 2 · · · x1,n+iy1,n √ 2 x1,2−iy1,2 √ 2 x2,2 · · · x1,n+iy1,n √ 2 . . . . . . ... . . . x1,n+iy1,n √ 2 xn,2−iyn,2 √ 2 · · · xn,n The eigenvalues of Xn have joint density given by P(λ1, . . . , λn) = 1 Zn,2 j<k |λj − λk|2e−2 4 n k=1 λ2 k