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• W2949546172 abstract "In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =sum_{i=1}^n c_i xi_i z^i$ and $tilde f =sum_{i=1}^n c_i tilde xi_i z^i$, where the $xi_i$ and $tilde xi_i$ are iid random variables that match moments to second order, the coefficients $c_i$ are deterministic, and the degree parameter $n$ is large. Our results show, under some light conditions on the coefficients $c_i$ and the tails of $xi_i, tilde xi_i$, that the correlation functions of the zeroes of $f$ and $tilde f$ are approximately the same. As an application, we give some answers to the classical question `How many zeroes of a random polynomials are real? for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions $f$ and $tilde f$ if their log magnitudes $log |f|, log|tilde f|$ are close in distribution, and if some non-concentration bounds are obeyed." @default.
• W2949546172 created "2019-06-27" @default.
• W2949546172 creator A5018669443 @default.
• W2949546172 creator A5066728260 @default.
• W2949546172 date "2013-07-16" @default.
• W2949546172 modified "2023-09-23" @default.
• W2949546172 title "Local universality of zeroes of random polynomials" @default.
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• W2949546172 doi "https://doi.org/10.48550/arxiv.1307.4357" @default.
• W2949546172 hasPublicationYear "2013" @default.
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