Almost sure weak convergence for the generalized orthogonal ensemble (Q5950966)

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 \(V(X)= \text{trace} v(X)/n\) defines a uniformly \(p\)-convex function on the real symmetric \(n\times n\) matrices \(X\) for some \(p\geq 2\). Then \(v(dX)= e^{-V(X)} dX/Z\) satisfies deviation and transportation inequalities analogous to those satisfied by Gaussian measure, but for the Schatten \(c^p\) norm. The map, that associates to each \(X\in M^s_n(\mathbb{R})\) its ordered eigenvalue sequence, induces from \(v\) 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 \(n\to\infty\).