Almost sure weak convergence for the generalized orthogonal ensemble

The generalized orthogonal ensemble satisfies isoperimetric inequalities an alogous to the Gaussian isoperimetric inequality, and an analogue of Wigner 's law. Let v be a continuous and even real function such that V(X) = trace v(X)/n defines a uniformly p-convex function on the real symmetric n x n m atrices X for some p greater than or equal to 2. Then v(dX) = e(-V(X))dX/Z satisfies deviation and transportation inequalities analogous to those sati sfied by Gaussian measure((6, 27)), but for the Schatten c(p) norm. The map , that associates to each X is an element of M-n(s) (R) its ordered eigenva lue sequence, induces from v a measure which satisfies similar inequalities , It follows from such concentration inequalities that the empirical distri bution of eigenvalues converges weakly almost surely to some non-random com pactly supported probability distribution as n --> infinity.