Abstract

In this chapter we combine the results from Chapter 3 about concentration of measure with the notion of equilibrium from Chapter 4, and prove convergence to equilibrium of empirical eigenvalue distributions of n × n matrices from suitable ensembles as n → ∞. We introduce various notions of convergence for eigenvalue ensembles from generalized orthogonal, unitary and symplectic ensembles. Using concentration inequalities from Chapter 3, we prove that the empirical eigenvalue distributions, from ensembles that have uniformly convex potentials, converge almost surely to their equilibrium distribution as the number of charges increases to infinity. Furthermore, we obtain the Marchenko–Pastur distribution as the limit of singular numbers of rectangular Gaussian matrices. To illustrate how concentration implies convergence, the chapter starts with the case of compact groups, where the equilibrium measure is simply normalized arclength on the circle.

Convergence to arclength

Suppose that n unit positive charges of strength β > 0 are placed upon a circular conductor of unit radius, and that the angles of the charges are 0 ≤ θ1 < θ2 < … < θn < 2π. Suppose that the θj are random, subject to the joint distribution

Then we would expect that the θj would tend to form a uniform distribution round the circle as n → ∞ since the uniform distribution appears to minimize the energy. We prove this for β = 2.