Limiting Root Distribution of Random Log-concave Polynomials

Abstract

We introduce two probabilistic models of random log-concave polynomials, the uniform model and the beta model, and study the asymptotic distribution of their zeros in the complex plane. In the uniform model, we show that the empirical root distribution converges to the uniform probability measure on the unit circle, placing the model in the same universality class as classical Kac polynomials. In contrast, in the beta model log-concavity is amplified through exponential scaling of the coefficients, leading to a new limiting distribution that is rotationally symmetric and absolutely continuous with respect to Lebesgue measure on the plane.