On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients

TL;DR

This paper studies the asymptotic distribution of roots of random polynomials with heavy-tailed coefficients, showing they concentrate on two circles with uniform argument distribution as degree increases.

Contribution

It establishes the limiting distribution of roots for polynomials with heavy-tailed i.i.d. coefficients, revealing concentration on two specific circles.

Findings

Roots concentrate on two circles as degree increases

Distribution of roots is uniform in argument in the limit

Radii depend on ratios of coefficients with maximum modulus

Abstract

Consider a random polynomial $G_n(z)=\xi_nz^n+...+\xi_1z+\xi_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\log(1+\log(1+|\xi_0|))$ has a slowly varying tail. Then the distribution of the complex roots of $G_n$ concentrates in probability, as $n\to\infty$, to two centered circles and is uniform in the argument as $n\to\infty$. The radii of the circles are $|\xi_0/\xi_\tau|^{1/\tau}$ and $|\xi_\tau/\xi_n|^{1/(n-\tau)}$, where $\xi_\tau$ denotes the coefficient with the maximum modulus.