Estimates for correlation functions of real roots for random polynomials and applications

A random polynomial is a polynomial whose coefficients are random variables. A major task in the theory of random polynomials is to examine how the real roots are distributed and correlated in situations where the degree of the polynomial is large. In this dissertation, we investigate two classes of random polynomials that have piqued the interest of researchers in probability theory and mathematical physics: elliptic polynomials and generalized Kac polynomials.

Regarding elliptic polynomials, we obtain asymptotic expansions for the variances of the number of real roots on intervals whose endpoints may vary based on the polynomial's degree. Additionally, we provide sharp estimates for the cumulants of these quantities. As applications, we can determine intervals on which the number of real roots satisfies a central limit theorem and a strong law of large numbers.

Our next objective is to compute the precise leading asymptotics of the variance of the number of real roots for generalized Kac polynomials whose coefficients have polynomial asymptotics. Examples of this class of random polynomials include Kac polynomials, hyperbolic polynomials, and any linear combinations of their derivatives. Before our work, such variance asymptotics had only been established for the Kac polynomials in the 1970s, thanks to Maslova's influential contribution. Our proof relies on novel asymptotic estimates for the real roots' two-point correlation function, which exposes geometric features in the distribution of real roots for these random polynomials. As a corollary, we establish asymptotic normality for the number of real roots of these random polynomials, extending and enhancing a related result of O. Nguyen and V. Vu.