ON THE DISTRIBUTION OF COMPLEX ROOTS OF RANDOM POLYNOMIALS WITH HEAVY-TAILED COEFFICIENTS?

摘要

Consider a random polynomial G_n(z) = ξ_nz~n + · · · + ξ_1z + ξ_0 with independent identically distributed complex-valued coefficients. Suppose that the distribution of log(1 + log(1 + |ξ0|)) has a slowly varying tail. Then the distribution of the complex roots of G_n concentrates in probability, as n→∞, to two centered circles and is uniform in the argument as n→∞. The radii of the circles are |ξ0/ξτ |~(1/τ) and |ξ_τ /ξ_n|~(1/(n?τ)), where ξ_τ denotes the coefficient with the maximum modulus. roots of a random polynomial, roots concentration, heavy-tailed coefficients