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|Title:||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  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 , 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 , 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.|
|Type of Material:||Princeton University Senior Theses|
|Appears in Collections:||Mathematics, 1934-2020|
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