Vladimir V. Ulyanov
The paper deals with different properties of polynomials in random elements: bounds for characteristics functionals of polynomials, stochastic generalization of the Vinogradov mean value theorem, characterization problem, bounds for probabilities to hit the balls. These results cover the cases where the random elements take values in finite as well as infinite dimensional Hilbert spaces.
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In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials. Various models of random polynomials are considered by introducing randomness through multiplying a factor with a random zero or removing a zero at random for a given sequence of deterministic polynomials. We also obtain similar results for random polynomials whose zeros are given by i.i.d. random variables. As an application, we show that these phenomenon a…
Let $p_n$ be the characteristic polynomial of an $n \times n$ random matrix drawn from one of the compact classical matrix groups. We show that the critical points of $p_n$ converge to the uniform distribution on the unit circle as $n$ tends to infinity. More generally, we show the same limit for a class of random polynomials whose roots lie on the unit circle. Our results extend the work of Pemantle-Rivin and Kabluchko to the setting where the roots are neither independent nor identically distributed.
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 polynomials, even when the coeffcients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.
We prove that with high probability, d+1 random Bernoulli polynomials in d variables of degree n (n goes to infinity) do not possess a common root.
We obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails and derive stable limit theorems for nonconventional sums of such polynomials
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 of real zeros. Almost all previous results require coefficients with zero mean, and it is non-trivial to extend these results to the general case. Our approach is based on a novel comparison principle that reduces the general situation to the mean-zero setting. As applications, we obtain new results for the Kac polynomials, hyperbolic random polynomials, th…
In this paper we study bounds for the total variation distance between two second degree polynomials in normal random variables provided that they essentially depend on at least three variables.
We show that the number of real roots of random trigonometric polynomials with i.i.d. coefficients, which are either bounded or satisfy the logarithmic Sobolev inequality, satisfies an exponential concentration of measure.
This article revisits the work by Ofer Zeitouni and Steve Zelditch on large deviations for the empirical measures of random orthogonal polynomials with i.i.d. Gaussian complex coefficients, and extends this result to real Gaussian coefficients. This article does not require any knowledge in geometry. For clarity, we focus on two classical cases: Kac polynomials and elliptic polynomials.
In this paper we find the asymptotic main term of the variance of the number of roots of Kostlan-Shub-Smale random polynomials and prove a central limit theorem for the number of roots as the degree goes to infinity.
The Distribution of Polynomials in Monotone Independent Elements
本文研究单调独立与无穷小单调独立元素中多项式的 $*$-分布，显式给出 $αab+βba$ 的分布，并应用于随机矩阵多项式的偏迹极限分布。
In this paper, using the method proposed by Dembo and Mukherjee [5], we obtain the persistence exponents of random Weyl polynomials in both cases: half nonnegative axis and the whole real axis. Our result is a confirmation to the predictions of Schehr and Majumdar [22].
We study the regularity of densities of distributions that are polynomial images of the standard Gaussian measure on $\mathbb{R}^n$. We assume that the degree of a polynomial is fixed and that each variable enters to a power bounded by another fixed number.
We present a concentration inequality for linear functionals of noncommutative polynomials in random matrices. Our hypotheses cover most standard ensembles, including Gaussian matrices, matrices with independent uniformly bounded entries and unitary or orthogonal matrices.
In this article we study asymptotic behavior of the probability that a random monic polynomial with integer coefficients is irreducible over the integers. We consider the cases where the coefficients grow together with the degree of the random polynomials. Our main result is a generalization of a theorem proved by Konyagin in 1999. We also generalize Hilbert's Irreducibility Theorem and present an analog of this result with centered Binomial distributed coefficients.
Real roots of random polynomials: asymptotics of the variance
本文针对系数呈多项式增长的一大类随机多项式，通过实根两点关联函数的新渐近估计，得到其实根个数方差的精确首项渐近性及渐近正态性，推广了既往仅针对Kac多项式的相关结果。
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 distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of ra…
We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $(2/π+ o(1))\log{n}$ expected real zeros in terms of the degree $n$. On the other hand, if the basis is given by Legendre (or more generally by Jacobi) polynomials, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the latter asymptotic relation holds universally for a large class of random orthogonal polynomials on the real line, and also give more …
We construct a coupling between the random walk composed of Lévy area increments from a $d$-dimensional Brownian motion and a random walk composed of quadratic polynomials of Gaussian random variables. This coupling construction is used to produce a new pathwise approximation scheme for stochastic differential equations in the preprint [Flint-Lyons-2015]. The coupling arguments of the present paper are based extensively on the recent coupling results of Davie concerning a multidimensional variant of the Komlós-Major-Tusnády theorem and Wasserstein estimates for polynomial perturbations of Gaus…
We consider a uniform distribution on the set $\mathcal{M}_k$ of moments of order $k \in \mathbb{N}$ corresponding to probability measures on the interval $[0,1]$. To each (random) vector of moments in $\mathcal{M}_{2n-1}$ we consider the corresponding uniquely determined monic (random) orthogonal polynomial of degree $n$ and study the asymptotic properties of its roots if $n \to \infty$.
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 moment of the number of roots and a Hermite expansion of this random variable.
Concentration of the number of real roots of random polynomials
本文系统研究了随机多项式实根数量的集中性问题，提出了一种初等方法，适用于高斯或非高斯系数的多种随机多项式模型，旨在获得近乎最优的界。
In this paper we consider random block matrices, which generalize the general beta ensembles, which were recently investigated by Dumitriu and Edelmann (2002, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density, which can be represented as the trace of an integral of densities of matrix measures corresponding to the C…
Maximum of the characteristic polynomial of random Jacobi matrices
本文研究了满足指数可积条件的随机雅可比矩阵特征多项式最大值的二级渐进行为，并通过三对角表示方法验证了Fydorov-Simm猜想。
Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us to study these numbers for a broad family of probability distributions. Our work also extends to perturbed systems.
The local time of random walks associated with Gegenbauer polynomials $P_n^{(α)}(x),\ x\in [-1,1]$ is studied in the recurrent case: $α\in\ [-\frac{1}{2},0]$. When $α$ is nonzero, the limit distribution is given in terms of a Mittag-Leffler distribution. The proof is based on a local limit theorem for the random walk associated with Gegenbauer polynomials. As a by-product, we derive the limit distribution of the local time of some particular birth and death Markov chains on $\bbN$.
This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the distribution of the roots of certain complex random polynomials. We discuss a recent result in this direction, which shows that the associated points in the sphere (via the stereographic projection) are surprisingly well-suited with respect to the minimal logarithmic energy on the sphere.
We provide a generalization and new proofs of the formulas of Collins, Hasebe and Sakuma for the spectrum of polynomials in cyclic monotone elements. This is applied to random matrices with discrete spectrum.
We consider about calculating $M$th moments of a given polynomial in free independent semicircular elements in free probability theory. By a naive approach, this calculation requires exponential time with respect to $M$. We explicitly give an algorithm for calculating them in polynomial time by rearranging Schützenberger's algorithm.
We give an explicit formula for the correlation functions of real zeros of a random polynomial with arbitrary independent continuously distributed coefficients.
The study of several naturally arising "nearest neighbours" random walks benefits from the study of the associated orthogonal polynomials and their orthogonality measure. I consider extensions of this approach to a larger class of random walks. This raises a number of open problems.
Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected number of roots in various sets. This is done for polynomials with coefficients that may be dependent, and need not have identical distributions. We also study random polynomials spanned by various deterministic bases.
Li and Wei (2009) studied the density of zeros of Gaussian harmonic polynomials with independent Gaussian coefficients. They derived a formula for the expected number of zeros of random harmonic polynomials as well as asymptotics for the case that the polynomials are drawn from the Kostlan ensemble. In this paper we extend their work to cover the case that the polynomials are drawn from the Weyl ensemble by deriving asymptotics for this class of harmonic polynomials.
The expected number of zeros of a random real polynomial of degree $N$ asymptotically equals $\frac{2}π\log N$. On the other hand, the average fraction of real zeros of a random trigonometric polynomial of increasing degree $N$ converges to not $0$ but to $1/\sqrt 3$. An average number of roots of a system of random trigonometric polynomials in several variables is equal to the mixed volume of some ellipsoids depending on the degrees of polynomials. Comparing this formula with Theorem BKK we prove that the phenomenon of nonzero fraction of real roots remains valid.
We study the supremum of some random Dirichlet polynomials and obtain sharp upper and lower bounds for supremum expectation that extend the optimal estimate of Halász-Queffélec and enable to cunstruct random polynomials with unusually small maxima. Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.
The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contains a wide range of random variables including standard Gaussian and all bounded random variables. We prove that for almost every sequence of random polynomials their normalized zero currents become equidistributed with respect to a deterministic extre…
We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems.
In this paper we establish a connection between the fluctuations of Wishart random matrices, shifted Chebyshev polynomials, and planar diagrams whose linear span form a basis for the irreducible representations of the annular Temperly-Lieb algebra.
We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular law for the empirical spectral distribution of these matrices when the 4th moments of their entries are controlled.
We consider the use of random walks as an approach to obtain connection coefficients for higher-order Bernoulli and Euler polynomials. In particular, we consider the cases of a $1$-dimensional linear reflected Brownian motion and of a $3$-dimensional Bessel process. Considering the successive hitting times of two, three, and four fixed levels by these random walks yields non-trivial identities that involve higher-order Bernoulli and Euler polynomials.
Motivated by the question of whether a random polynomial with integer coefficients is likely to be irreducible, we study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers, which is equivalent to having a low-degree algebraic root. It is known in certain cases that random polynomials with integer coefficients are very likely to be irreducible, and our project can be viewed as part of the general program of testing whether this is a universal behavior exhibited by many random polynomial models. Our main result shows that pointwise delocal…
A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived and the eigenvalues are localised. Three instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product $HX$ of a fixed diagonal matrix $H$ and the Wigner matrix $X$ and a special matrix polynomial. The results are illustrated with various examples and numerical simulations.
Probabilistic degenerate Fubini polynomials associated with random variables
本文研究退化Fubini多项式及r阶退化Fubini多项式的概率版本，推导了其性质、显式表达、恒等式与递推关系。
A complex harmonic polynomial is the sum of a complex polynomial and a conjugated complex polynomial, of degrees $n$ and $m$ respectively. Li and Wei (2009) presented a formula for the expected number of zeros of a random harmonic polynomial in $\mathbb{C}$. In this paper we prove that if $m$ is a fixed number, this expectation is asymptotically $n$ as $n\rightarrow\infty$, and if $m=n$ we find a lower and upper bound of order $n\log n$ for this expectation for sufficiently large $n$.
In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by a multi-circular weight function $Q$ satisfying suitable smoothness and growth conditions. In complex dimension $m\geq3$, we prove that $\Bbb{E}[(\log(1+|a_j|))^m]<\infty$ is a necessary and sufficient condition for normalized zero currents of random polynomials to be almost surely asymptotic to the (deterministic) extremal current $dd^cV_{Q}.$ In addition, …
We show that almost all the zeros of any finite linear combination of independent characteristic polynomials of random unitary matrices lie on the unit circle. This result is the random matrix analogue of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of $L$-functions, thus providing further evidence for the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of $L$-functions on the critical line.
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 Erdos 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.
We show that there is no algorithm which, provided a polynomial number of random points uniformly distributed over a convex body in R^n, can approximate the volume of the body up to a constant factor with high probability.
We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric polynomials has, on average, many real roots. In the case that the coefficients of a real trigonometric polynomial are independently and identically distributed, but with no other assumptions on the distribution, the expected fraction of real zeros is at least one-half. This result is best possible.
We study the supremum of random Dirichlet polynomials $D_N(t)=\sum_{n=1}^N\varepsilon_n d(n) n^{- s}$, where $(\varepsilon_n)$ is a sequence of independent Rademacher random variables, and $ d $ is a sub-multiplicative function. The approach is gaussian and entirely based on comparison properties of Gaussian processes, with no use of the metric entropy method.
It is well known that the expected number of real zeros of a random cosine polynomial $ V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2π) $, with the $ a_j $ being standard Gaussian i.i.d. random variables is asymptotically $ 2n / \sqrt{3} $. On the other hand, some of the previous works on the random cosine polynomials with dependent coefficients show that such polynomials have at least $ 2n / \sqrt{3} $ expected real zeros lying in one period. In this paper we investigate two classes of random cosine polynomials with pairwise equal blocks of coefficients. First, we prove that a rando…
We study the asymptotic distribution of zeros for the random polynomials $P_n(z) = \sum_{k=0}^n A_k B_k(z)$, where $\{A_k\}_{k=0}^{\infty}$ are non-trivial i.i.d. complex random variables. Polynomials $\{B_k\}_{k=0}^{\infty}$ are deterministic, and are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain $G$ bounded by an analytic curve. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$ if and only if $\mathbb{E}[\log^+|A_0|]<\infty$.
In this note we study the number of real roots of a wide class of random orthogonal polynomials with gaussian coefficients. Using the method of Wiener Chaos we show that the fluctuation in the bulk is asymptotically gaussian, even when the local correlations are not necessarily the same.
We prove that with "high probability" a random Kostlan polynomial in $n+1$ many variables and of degree $d$ can be approximated by a polynomial of "low degree" without changing the topology of its zero set on the sphere $S^n$. The dependence between the "low degree" of the approximation and the "high probability" is quantitative: for example, with overwhelming probability the zero set of a Kostlan polynomial of degree $d$ is isotopic to the zero set of a polynomial of degree $O(\sqrt{d \log d})$. The proof is based on a probabilistic study of the size of $C^1$-stable neighborhoods of Kostlan p…
In this work we design a general method for proving moment inequalities for polynomials of independent random variables. Our method works for a wide range of random variables including Gaussian, Boolean, exponential, Poisson and many others. We apply our method to derive general concentration inequalities for polynomials of independent random variables. We show that our method implies concentration inequalities for some previously open problems, e.g. permanent of a random symmetric matrices. We show that our concentration inequality is stronger than the well-known concentration inequality due …
Irreducibility of polynomials with random multiplicative coefficients revisited
本文针对取值于{±1}的随机完全乘法函数构成的多项式，证明其不可约概率的上极限随次数趋于无穷时等于1，推进了随机乘法系数多项式不可约性的研究。
It has been shown recently that the limit moments of $W(n)=B(n)B^{*}(n)$, where B(n) is a product of $p$ independent rectangular random matrices, are certain homogenous polynomials in the asymptotic dimensions of these matrices. Using the combinatorics of noncrossing partitions, we explicitly determine these polynomials and show that they are closely related to polynomials which can be viewed as multivariate Fuss-Narayana polynomials. Using this result, we compute the moments of the n-fold free multiplicative convolution of Marchenko-Pastur distributions with arbitrary shape parameters.
We point out a simple criterion for convergence of polynomials to a concrete entire function in the Laguerre-Pólya ($\mathcal{LP}$) class (of all functions arising as uniform limits of polynomials with only real roots). We then use this to show that any random $\mathcal{LP}$ function can be obtained as the uniform limit of rescaled characteristic polynomials of principal submatrices of an infinite unitarily invariant random Hermitian matrix. Conversely, the rescaled characteristic polynomials of principal submatrices of any infinite random unitarily invariant Hermitian matrix converge uniforml…
We derive a large deviation principle for the empirical measure of zeros of random polynomials with i.i.d. exponential coefficients.
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