Critical Points of Random Polynomials and Characteristic Polynomials of Random Matrices

Let p(n) be the characteristic polynomial of an n x n random matrix drawn from one of the compact classical matrix groups. We show that the critical points of pn converge to the uniform distribution on the unit circle as n tends to infinity. More generally, we show the same limit for a class of random polynomials whose roots lie on the unit circle. Our results extend the work of Pemantle-Rivin [30] and Kabluchko [19] to the setting where the roots are neither independent nor identically distributed.