The zeros of random polynomials cluster uniformly near the unit circle
نویسندگان
چکیده
In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdős and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.
منابع مشابه
The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle
We consider polynomials on the unit circle defined by the recurrence relation Φk+1(z) = zΦk(z)− αkΦk(z) k ≥ 0, Φ0 = 1 For each n we take α0, α1, . . . , αn−2 i.i.d. random variables distributed uniformly in a disk of radius r < 1 and αn−1 another random variable independent of the previous ones and distributed uniformly on the unit circle. The previous recurrence relation gives a sequence of ra...
متن کاملAsymptotic zero distribution of random polynomials spanned by general bases
Zeros of Kac polynomials spanned by monomials with i.i.d. random coefficients are asymptotically uniformly distributed near the unit circumference. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal...
متن کاملZero Distribution of Random Polynomials
We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures, and to quantitative results on the expected number of zeros in various sets. In the simplest case of Kac polynomials, given by the linear combinations of monomials with i.i.d. random coefficients, it is well known that th...
متن کاملOn the Distribution of Zeros of Polynomials Orthogonal on the Unit Circle
Let { n} be a system of monic polynomials orthogonal on the unit circle with respect to a positive Borel measure p.. It is shown that under .fairly mild conditions on the reflection coefficients, n(O), the zeros of a subsequence of { n} are asymptotically uniformly distributed on some circle. The radius of this circle can be found using a Cauchy-Hadamard-type formula. We also character...
متن کاملUniversités de Paris 6 & Paris 7 - CNRS ( UMR 7599 ) PRÉPUBLICATIONS DU LABORATOIRE DE PROBABILITÉS & MODÈLES ALÉATOIRES
Given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. In particular, we do not assume independence or equidistribution of the coefficients of the polynomial. We apply this result to various problems in both random and deterministic sequences of polyno-mials, including ...
متن کامل