Limiting Distribution of the Complex Roots of Random Polynomials

Abstract

In this thesis, we follow the proof of a general result of Kabluchko and Zaporozhets in [8] about the limiting distribution of the complex roots of a (finite or infinite) random polynomial indexed by a natural number n, as n goes to infinity, when the coefficients of the polynomial satisfy certain conditions. We briefly discuss a recent and related result by Bloom and Dauvergne in [1], which gives a stronger sense of convergence for polynomials of degree n. Next, we investigate the limiting distributions for a class of polynomials considered by Schehr and Majumdar in [11], in which the asymptotics for the expected number of real roots exhibit phase transitions along a parameter \alpha. We provide a conjecture for the limiting distributions of this class, using a heuristic argument based on the methods in the result of Kabluchko and Zaporozhets, and find that, if the conjecture is correct, phase transitions occur in the limiting distributions, as well.