相关论文: Real roots of random polynomials: asymptotics of t…

In this paper, we study the number of real roots of random trigonometric polynomials with iid coefficients. When the coefficients have zero mean, unit variance and some finite high moments, we show that the variance of the number of real…

We obtain the asymptotic variance, as the degree goes to infinity, of the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size. Our main tools are the Kac-Rice formula for the second factorial…

We study the fluctuations of the number of real roots of random polynomials with independent, nonzero-mean coefficients. Such non-centered ensembles arise naturally in signal-plus-noise models and in random perturbations of deterministic…

This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number…

For each $\alpha \in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum_{k=0}^n c_k \varepsilon_k z^k$ is $n^{\alpha +…

The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood-Offord and Kac in the 1940s. The main task here is to determine the limiting…

We identify the scaling region of a width O(n^{-1}) in the vicinity of the accumulation points $t=\pm 1$ of the real roots of a random Kac-like polynomial of large degree n. We argue that the density of the real roots in this region tends…

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on…

Addressing a problem posed by W. Li and A. Wei (2009), we investigate the average number of (complex) zeros of a random harmonic polynomial $p(z) + \overline{q(z)}$ sampled from the Kac ensemble, i.e., where the coefficients are independent…

In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coeffcients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these…

We consider random trigonometric polynomials with general dependent coefficients. We show that under mild hypotheses on the structure of dependence, the asymptotics as the degree goes to infinity of the expected number of real zeros…

We compute the variance asymptotics for the number of real zeros of trigonometric polynomials with random dependent Gaussian coefficients and show that under mild conditions, the asymptotic behavior is the same as in the independent…

We consider random orthonormal polynomials $$ F_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, \dots, $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep)$ moments, and…

It is well known that a random cosine polynomial $ V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2 \pi) $, with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables…

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence…

We obtain a Central Limit Theorem for the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size as the degree goes to infinity. A study of the asymptotic variance of the number of roots is…

Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the…

A new short clear proof of the asymptotics for the number $c_n$ of real roots of the Bernoulli polynomials $B_n(x)$, as well as for the maximal root $y_n$: $$y_n=\frac{n}{2\pi e}+\frac{\ln(n)}{4\pi e}+O(1)\quad\text{and}\quad…

In this paper, we investigate the number of real zeros of random Weyl polynomials of degree \(n \to \infty\) with general coefficient distributions. Motivated by the results of arXiv:1409.4128 and arXiv:1402.4628 as well as arXiv:1711.03316…

We consider random polynomials $p_n(x)=\xi_0+\xi_1+\dots+\xi_n x^n$ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded $(2+\epsilon)^{th}$ moment (for some $\epsilon>0$), also known as…