Universal large deviations for Kac polynomials

Raphaël Butez,Ofer Zeitouni

doi:10.1214/16-ecp33

Abstract

Highlights

We prove the universality of the large deviations principle for the empirical measures of zeros of random polynomials whose coefficients are i.i.d. random variables possessing a density with respect to the Lebesgue measure on C, R or R+, under the assumption that the density does not vanish too fast at zero and decays at least as exp −|x|ρ, ρ > 0, at infinity
Where a0, . . . , an are i.i.d. random variables and z1, . . . , zn are the complex zeros of Pn. (Such random polynomials are often referred to as Kac polynomials.) There is a rich literature about the behavior of the zeros of Pn and we refer to [TV15] for a nice recent review of the subject
In case the coefficients are i.i.d. standard complex Gaussian random variables, Zeitouni and Zelditch[1] proved in [ZZ10] that the sequence of empirical measures of zeros satisfies the large deviations principle (LDP) in M1(C) with speed n2 and good rate function IC defined by IC(μ) = −

Summary

Introduction

We prove the universality of the large deviations principle for the empirical measures of zeros of random polynomials whose coefficients are i.i.d. random variables possessing a density with respect to the Lebesgue measure on C, R or R+, under the assumption that the density does not vanish too fast at zero and decays at least as exp −|x|ρ, ρ > 0, at infinity. In case the coefficients (ai) are i.i.d. standard complex Gaussian random variables, Zeitouni and Zelditch[1] proved in [ZZ10] that the sequence of empirical measures of zeros (which we denote by μCn for this model) satisfies the large deviations principle (LDP) in M1(C) with speed n2 and good rate function IC defined by IC(μ) = − Gaussians (ai): the empirical measure of zeros, denoted μRn for that model, satisfies the LDP in M1(C) with speed n2 and good rate function IR defined by IR(μ) =

Results

Conclusion