The largest root of random Kac polynomials is heavy tailed

Raphaël Butez

doi:10.1214/18-ecp114

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 deviations principle for the empirical measure is universal. This is in sharp contrast with random matrix theory where the large deviations 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 Virág [15] to obtain explicit formulas for the limiting objects.

Highlights

Consider a random polynomial of the form: n nPn(z) = akzk = an (z − zk(n))k=0 k=1 where a0, . . . , an are i.i.d. random variables and z1(n), . . . , zn(n) are the complex zeros of Pn
K=0 k=1 where a0, . . . , an are i.i.d. random variables and z1(n), . . . , zn(n) are the complex zeros of Pn. These polynomials are often called Kac polynomials. The zeros of these polynomials are known to concentrate on the unit circle of C as their degree tends to infinity under some moment condition on the coefficients
This universal behavior has been studied by many authors since the work of Sparo and Shur [ŠŠ62] and we refer to the book [BRS86] for more precise information on the history of the topic

Summary

Introduction

Consider a random polynomial of the form:. k=0 k=1 where a0, . . . , an are i.i.d. random variables and z1(n), . . . , zn(n) are the complex zeros of Pn. For Ginibre random matrices or Kac random polynomials, the empirical measures converge towards a deterministic measure and the explicit distribution of the eigenvalues (or zeros) can be computed when the coefficients are Gaussian (real or complex). If we look at the behavior of the confining term in each variable for random polynomials, we see that it grows at infinity like log(|z|) while the confining term for Ginibre is V (z) = |z|2/2 We believe that it is a general fact for Coulomb gases: when the confining term is of order log |z| at infinity, the largest particle has a heavy tail and when the confining term is stronger than logarithm, the largest particle should converge towards the edge of the limiting distribution. We use this inequality to the non negative variable

We deduce that

This implies that