The distribution of the zeroes of random trigonometric polynomials
Andrew Granville, Igor Wigman
备注：
51 pages. We cut the size of the paper to better suit publication. In particular, all the results of empirical experiments were cut off. Some standard results in probability and stochastic processes were also omitted. Numerous typos and mistakes were corrected following the suggestions of referees. This paper was accepted for publication in the American Journal of Mathematics.
We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that $\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.
Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $ξ_0,ξ_1,\ldots,ξ_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\to\infty$, with high probability there is a critical point of $Q_n$ which is very close to $ξ_0$. We localize the position of this critical point by proving that the difference between $ξ_0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(ξ_0))$ and variance of order $\log n …
We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for zeroes lying on any curve in the complex plane, which is the graph of a Lipschitz function in polar coordinates. The proof is based on the classical Turán lemma.
In this sequel to Part-I, we present a different approach to bounding the expected number of real zeroes of random polynomials with real independent identically distributed coefficients or more generally, exchangeable coefficients. We show that the mean number of real zeroes does not grow faster than the logarithm of the degree. The main ingredients of our approach are Descartes' rule of signs and a new anti-concentration inequality for the symmetric group. This paper can be read independently of part-I in this series.
In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =\sum_{i=1}^n c_i ξ_i z^i$ and $\tilde f =\sum_{i=1}^n c_i \tilde ξ_i z^i$, where the $ξ_i$ and $\tilde ξ_i$ are iid random variables that match moments to second order, the coefficients $c_i$ are deterministic, and the degree parameter $n$ is large. Our results show, under some light conditions on the coefficients $c_i$ and the tails of $ξ_i, \tilde ξ_i$, that the correlation funct…
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.
We prove that the minimum of the modulus of a random trigonometric polynomial with Gaussian coefficients, properly normalized, has limiting exponential distribution.
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…
Let $μ$ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots, z_n$ be independent random variables, $z_i \sim μ$, and consider the random polynomial $$ p_n(z) = \prod_{k=1}^{n}{(z - z_k)}.$$ We determine the asymptotic distribution of $\left\{z \in \mathbb{C}: p_n(z) = p_n(0)\right\}$. In particular, if $μ$ is radial around the origin, then those solutions are also distributed according to $μ$ as $n \rightarrow \infty$. Generally, the distribution of the solutions will reproduce parts of $μ$ and condense another part on curves. We…
The average value of the number of intersection points of two plane curves given by trigonometrical polynomial maps of degree $N$ and bounded $L_2$- or $W_2^r$- norms is calculated
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 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 framework. In fact our proof goes beyond this framework and makes explicit the variance asymptotics of various models of random Gaussian polynomials. Though we use the Kac--Rice formula, we do not use the explicit closed formula for the second moment of the number of zeros, but we rather rely on intrinsic properties of the Kac--Rice density.
We prove a Central Limit Theorem for the number of zeros of random trigonometric polynomials of the form $K^{-1/2}\sum_{n=1}^{K} a_n\cos(nt)$, being $(a_n)_n$ independent standard Gaussian random variables. In particular, we prove the conjecture by Farahmand, Granville & Wigman that the variance is equivalent to $V^2K$, $0<V^2<\infty$, as $K\to\infty$. % The case of stationary trigonometric polynomials was studied by Granville & Wigman and by Aza\"\is & León. Our approach is based on the Hermite/Wiener-Chaos decomposition for square-integrable functionals of a Gaussian process and on Rice Form…
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.
It has been shown in a recent work by Yakir-Zeitouni that the minimum modulus of random trigonometric polynomials with Gaussian coefficients has a limiting exponential distribution. We show this is a universal phenomenon. Our approach relates the joint distribution of small values of the polynomial at a fixed number $m$ of points on the circle to the distribution of a certain random walk in a $4m$-dimensional phase space. Under Diophantine approximation conditions on the angles, we obtain strong small ball estimates and a local central limit theorem for the distribution of the walk.
We study the locations of complex zeroes of independence polynomials of bounded degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer's result on the optimal zero-free disk, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $Δ$ have a zero-free disk almo…
We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials $$ X_n(t)=u+\frac{1}{\sqrt{n}}\sum_{k=1}^n (A_k\cos(kt)+B_k\sin(kt)), \quad t\in [0,2π],\quad u\in\mathbb{R} $$ whose coefficients $A_k, B_k$, $k\in\mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a, b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]\subseteq [0,2π]$, we prove that $$ \lim_{n\rightarrow\infty} \frac{\mathbb{E} N_n[a,b]}{n}=\frac{b-a}{π\sqrt{3}} e^{-\frac{u^2}{2}}. $$
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 roots is asymptotically linear in terms of the expectation; furthermore, the multiplicative constant in this linear relationship depends only on the kurtosis of the common distribution of the polynomial's coefficients. This result is in sharp contrast to the classical Kac polynomials whose corresponding variance depends only on the first two moments. Our resul…
The large degree asymptotics of the expected number of real zeros of a random trigonometric polynomial $$ T_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) + b_j \sin (j x), \ x \in (0,2π), $$ with i.i.d. real-valued standard Gaussian coefficients is known to be $ 2n / \sqrt{3} $. In this article, we consider quite a different and extreme setting on the set of the coefficients of $ T_n $. We show that a random trigonometric polynomial of degree $ n $ with $ \ell $-periodic i.i.d. Gaussian coefficients is expected to have significantly more real zeros compared to the classical case with i.i.d. Gaussian …
We give a precise measure of the rate at which repeated differentiation of a random trigonometric polynomial causes the roots of the function to approach equal spacing. This can be viewed as a toy model of crystallization in one dimension. In particular we determine the asymptotics of the distribution of the roots around the crystalline configuration and find that the distribution is not Gaussian.
In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein-1 distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous…
Motivated by problems arising in random sampling of trigonometric polynomials, we derive exponential inequalities for the operator norm of the difference between the sample second moment matrix $n^{-1}U^*U$ and its expectation where $U$ is a complex random $n\times D$ matrix with independent rows. These results immediately imply deviation inequalities for the largest (smallest) eigenvalues of the sample second moment matrix, which in turn lead to results on the condition number of the sample second moment matrix. We also show that trigonometric polynomials in several variables can be learned f…
We consider random trigonometric polynomials of the form \[ f_n(t):=\sum_{1\le k \le n} a_{k} \cos(kt) + b_{k} \sin(kt), \] whose entries $(a_{k})_{k\ge 1}$ and $(b_{k})_{k\ge 1}$ are given by two independent stationary Gaussian processes with the same correlation function $ρ$. Under mild assumptions on the spectral function $ψ_ρ$ associated with $ρ$, we prove that the expectation of the number $N_n([0,2π])$ of real roots of $f_n$ in the interval $[0,2π]$ satisfies \[ \lim_{n \to +\infty} \frac{\mathbb E\left [N_n([0,2π])\right]}{n} = \frac{2}{\sqrt{3}}. \] The latter result not only covers th…
Let $X_N$ be a random trigonometric polynomial of degree $N$ with iid coefficients and let $Z_N(I)$ denote the (random) number of its zeros lying in the compact interval $I\subset\mathbb{R}$. Recently, a number of important advances were made in the understanding of the asymptotic behaviour of $Z_N(I)$ as $N\to\infty$, in the case of standard Gaussian coefficients. The main theorem of the present paper is a universality result, that states that the limit of $Z_N(I)$ does not really depend on the exact distribution of the coefficients of $X_N$. More precisely, assuming that these latter are iid…
Stochastic dynamical systems often contain nonlinearities which make it hard to compute probability density functions or statistical moments of these systems. For the moment computations, nonlinearities in the dynamics lead to unclosed moment dynamics; in particular, the time evolution of a moment of a specific order may depend both on moments of order higher than it and on some nonlinear function of other moments. The moment closure techniques are used to find an approximate, close system of equations the moment dynamics. In this work, we extend a moment closure technique based on derivative …
We investigate the mean number of real zeros over an interval $[a,b]$ of a random trigonometric polynomial of the form $\sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$ where the coefficients are i.i.d. random variables. Under mild assumptions on the law of the entries, we prove that this mean number is asymptotically equivalent to $\frac{n(b-a)}{π\sqrt{3}}$ as $n$ goes to infinity, as in the known case of standard Gaussian coefficients. Our principal requirement is a new Cramer type condition on the characteristic function of the entries which does not only hold for all continuous distributions but al…
Consider a random trigonometric polynomial $X_n: \mathbb R \to \mathbb R$ of the form $$ X_n(t) = \sum_{k=1}^n \left( ξ_k \sin (kt) + η_k \cos (kt)\right), $$ where $(ξ_1,η_1),(ξ_2,η_2),\ldots$ are independent identically distributed bivariate real random vectors with zero mean and unit covariance matrix. Let $(s_n)_{n\in\mathbb N}$ be any sequence of real numbers. We prove that as $n\to\infty$, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+ b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$ of a stationary, zero-mean Gaussian process with correla…
We consider random trigonometric polynomials of the form \[ f_n(x,y)=\sum_{1\le k,l \le n} a_{k,l} \cos(kx) \cos(ly), \] where the entries $(a_{k,l})_{k,l\ge 1}$ are i.i.d. random variables that are centered with unit variance. We investigate the length $\ell_K(f_n)$ of the nodal set $Z_K(f_n)$ of the zeros of $f_n$ that belong to a compact set $K \subset \mathbb R^2$. We first establish a local universality result, namely we prove that, as $n$ goes to infinity, the sequence of random variables $n\, \ell_{K/n}(f_n)$ converges in distribution to a universal limit which does not depend on the pa…
We consider random trigonometric polynomials of the form \[ f_n(t):=\frac{1}{\sqrt{n}} \sum_{k=1}^{n}a_k \cos(k t)+b_k \sin(k t), \] where $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent stationary Gaussian processes with the same correlation function $ρ: k \mapsto \cos(kα)$, with $α\geq 0$. We show that the asymptotics of the expected number of real zeros differ from the universal one $\frac{2}{\sqrt{3}}$, holding in the case of independent or weakly dependent coefficients. More precisely, for all $\varepsilon>0$, for all $\ell \in (\sqrt{2},2]$, there exists $α\geq 0$ and $n\geq …
