The largest root of random Kac polynomials is heavy tailed

TL;DR

This paper investigates the behavior of the largest and smallest roots of random Kac polynomials, revealing their non-universal, heavy-tailed nature and contrasting this with universal properties in random matrix theory.

Contribution

It demonstrates the non-universality of extremal roots in Kac polynomials and characterizes their heavy-tailed distribution, contrasting with known universal behaviors in related fields.

Findings

Largest root modulus is heavy tailed with finite moments bounded by coefficient distribution.

Roots with modulus less than one converge to a limit point process.

Explicit formulas obtained for complex Gaussian coefficients case.

Abstract

We prove that the largest and smallest root in modulus of random Kac polynomials have a non-universal behavior. They do not converge towards the edge of the support of the limiting distribution of the zeros. This non-universality is surprising as the large deviation principle for the empirical measure is universal. This is in sharp contrast with random matrix theory where the large deviation principle is non-universal but the fluctuations of the largest eigenvalue are universal. We show that the modulus of the largest zero is heavy tailed, with a number of finite moments bounded from above by the behavior at the origin of the distribution of the coefficients. We also prove that the random process of the roots of modulus smaller than one converges towards a limit point process. Finally, in the case of complex Gaussian coefficients, we use the work of Peres and Vir{\'a}g [PV05] to obtain explicit formulas for the limiting objects.