Sums of random polynomials with independent roots

Abstract

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let p and q be two independent random polynomials of degree n, whose roots are chosen independently from the probability measures μ and ν in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum p+q as n tends to infinity. The limiting distribution can be described by its logarithmic potential, which we show is the pointwise maximum of the logarithmic potentials of μ and ν. More generally, we consider the sum of m independent degree n random polynomials when m is fixed and n tends to infinity. Our results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials.