Limiting Root Distribution of Random Log-concave Polynomials

TL;DR

This paper studies the asymptotic distribution of zeros of two models of random log-concave polynomials, revealing universal and novel limiting behaviors in the complex plane.

Contribution

It introduces the uniform and beta models of random log-concave polynomials and characterizes their asymptotic root distributions, including a new distribution for the beta model.

Findings

Uniform model roots converge to the unit circle

Beta model roots have a new rotationally symmetric distribution

Models belong to different universality classes

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.