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The largest root of random Kac polynomials is heavy tailed

Butez, Raphaël (2018), The largest root of random Kac polynomials is heavy tailed, Electronic Communications in Probability, 23, p. n°20. 10.1214/18-ECP114

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-01504174
Date
2018
Journal name
Electronic Communications in Probability
Volume
23
Pages
n°20
Publication identifier
10.1214/18-ECP114
Metadata
Show full item record
Author(s)
Butez, Raphaël
Abstract (EN)
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 Virág [PV05] to obtain explicit formulas for the limiting objects.
Subjects / Keywords
Random polynomials

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