Convergence of roots of expected characteristic polynomials of unitary ensembles to the equilibrium measure

Archit Kulkarni University of California, Berkeley (UC Berkeley) Mathematics

Let X_n be an n x n self-adjoint random matrix with probability density function C exp(-n Tr(V(X))), where V(x) is a polynomial of even degree with positive leading coefficient. Then the expected characteristic polynomials of X_n for varying n are the orthogonal polynomials with respect to the measure e^(-V(x)) dx. As n approaches infinity, the normalized counting measure on the roots of these polynomials (appropriately scaled) converges weakly to the so-called equilibrium measure for V, an object studied in potential theory. The equilibrium measure is also the weak limit of the mean empirical spectral distribution of X_n as n goes to infinity. As a special case of this, if V(x) = x^2/2, then X_n is drawn from the Gaussian Unitary Ensemble, the expected characteristic polynomials are the Hermite polynomials, and the equilibrium measure for V is the Wigner semicircle law.