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

Abstract

Consider a random polynomial $G_n(z)={\xi }_n{z}^n+...+{\xi }_{1}z+{\xi }_0$ 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.

8 pages