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

Consider a random polynomial $G_n(z)=ξ_nz^n+...+ξ_1z+ξ_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\log(1+\log(1+|ξ_0|))$ 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 $|ξ_0/ξ_τ|^{1/τ}$ and $|ξ_τ/ξ_n|^{1/(n-τ)}$, where $ξ_τ$ denotes the coefficient with the maximum modulus.

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