Almost Sure Weak Convergence for the Generalized Orthogonal Ensemble

摘要

The generalized orthogonal ensemble satisfies isoperimetric inequalities analogous 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)=tracev(X) defines a uniformly p-convex function on the real symmetric n×n matrices X for some p≥2. Then ν(dX)=e −V(X) dX/Z satisfies deviation and transportation inequalities analogous to those satisfied by Gaussian measure(6, 27), but for the Schatten c p norm. The map, that associates to each X∈M s n (ℝ) its ordered eigenvalue sequence, induces from ν a measure which satisfies similar inequalities. It follows from such concentration inequalities that the empirical distribution of eigenvalues converges weakly almost surely to some non-random compactly supported probability distribution as n→∞.