相关论文: A strong law of large numbers for real roots of ra…

Let $P_{n}(x)= \sum_{i=0}^n \xi_i x^i$ be a Kac random polynomial where the coefficients $\xi_i$ are iid copies of a given random variable $\xi$. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads…

We study concentration inequalities for the number of real roots of the classical Kac polynomials $$f_{n} (x) = \sum_{i=0}^n \xi_i x^i$$ where $\xi_i$ are independent random variables with mean 0, variance 1, and uniformly bounded…

Let $P_n(x) = \sum_{k=0}^{n} \xi_k x^k$ be a Kac random polynomial, where the coefficients $\xi_k$ are i.i.d.\ copies of a given random variable $\xi$. Based on numerical experiments, it has been conjectured that if $\xi$ has mean zero,…

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…

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 +…

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…

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…

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…

We compute the precise leading asymptotics of the variance of the number of real roots for a large class of random polynomials, where the random coefficients have polynomial growth. Our results apply to many classical ensembles, including…

We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and…

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables in $\mathbb{C}$ chosen from a probability measure $\mu$ and define the random polynomial $$ P_n(z)=(z-X_1)\ldots(z-X_n)\,. $$ We show that for any sequence $k…

Let $f_n$ be a random polynomial of degree $n$ with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of $f_n$ cluster uniformly around the unit circle as $n$ grows large. We give a simple and…

In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)=\sum_{k=1}^n Y_{k,1} \cos(kt)+Y_{k,2}\sin(kt)$ for a given sequence of i.i.d. random variables $\{Y_{k,1},Y_{k,2}\}_{k\ge 1}$ which are centered…

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…

Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the…

The Kac polynomial $$f_n(x) = \sum_{i=0}^{n} \xi_i x^i$$ with independent coefficients of variance 1 is one of the most studied models of random polynomials. It is well-known that the empirical measure of the roots converges to the uniform…

We study the number of real roots of a Kostlan random polynomial of degree $d$ in one variable. More generally, we are interested in the distribution of the counting measure of the set of real roots of such a polynomial. We compute the…

We introduce a family of real random polynomials of degree n whose coefficients a_k are symmetric independent Gaussian variables with variance <a_k^2> = e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly the mean number of…

In this paper, we study the root distribution of some univariate polynomials $W_n(z)$ satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the…

We opt to study the convergence of maximal real roots of certain Fibonacci-type polynomials given by $G_n=x^kG_{n-1}+G_{n-2}$. The special cases $k=1$ and $k=2$ are found in [4] and [7], respectively.