Browsing by Subject "random polynomials"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC
    (Elsevier, 2019-01) Aljubran, Hanan; Yattselev, Maxim L.; Mathematical Sciences, School of Science
    Let {φi}∞i=0 be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say En(μ), of random polynomials Pn(z):=∑i=0nηiφi(z), where η0,…,ηn are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that En(|dξ|) admits an asymptotic expansion of the form En(|dξ|)∼2πlog(n+1)+∑p=0∞Ap(n+1)−p (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon-Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case En(μ) admits an analogous expansion with coefficients the Ap depending on the measure μ for p≥1 (the leading order term and A0 remain the same).
  • Thumbnail Image
    Item
    On Random Polynomials Spanned by OPUC
    (2020-12) Aljubran, Hanan; Yattselev, Maxim; Bleher, Pavel; Mukhin, Evgeny; Roeder, Roland
    We consider the behavior of zeros of random polynomials of the from \begin{equation*} P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z) \end{equation*} as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.
  • Thumbnail Image
    Item
    The reciprocal Mahler ensembles of random polynomials
    (World Scientific, 2018) Sinclair, Christopher D.; Yattselev, Maxim L.; Mathematical Sciences, School of Science
    We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree N whose Mahler measure is bounded by a constant. After a change of variables, this reduces to a generalization of Ginibre’s complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval [−2,2] on the real axis in the complex plane. In the complex (real) case, the random roots form a determinantal (Pfaffian) point process, and in both cases, the empirical measure on roots converges weakly to the arcsine distribution supported on [−2,2]. Outside this region, the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from [−2,2]. These kernels as well as the scaling limits for the kernels in the bulk (−2,2) and at the endpoints {−2,2} are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.