[Paper Review] Central limit theorems for the real zeros of Weyl polynomials

Yen Do, Van Vu|arXiv (Cornell University)|Jul 28, 2017

Geometry and complex manifolds18 references3 citations

TL;DR

This paper establishes the central limit theorem (CLT) for the number of real zeros of Weyl polynomials, a class of random polynomials with independent Gaussian coefficients scaled by $1/\sqrt{k!}$. Using novel estimates for correlation functions and a comparison argument based on local laws and root repulsion, the authors prove that the normalized number of real zeros converges to a normal distribution, with variance asymptotically $ (2K + o(1))\sqrt{n} $, where $ K \approx 0.18198 $ is an explicit constant.

ABSTRACT

We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=xi_0 + xi_1 x+ ... + xi_n (n!)^{(-1/2)} x^n$, where $xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws and repulsion properties of these real roots.

Motivation & Objective

• To establish the central limit theorem (CLT) for the number of real zeros of Weyl polynomials, a key class of random polynomials with Gaussian coefficients.
• To extend the CLT beyond Kac and Kostlan-Shub-Smale polynomials to the Weyl polynomial case, which exhibits different scaling behavior.
• To develop a new method for proving CLTs in random polynomial theory that applies to general linear statistics of real zeros, not just the count.
• To quantify the asymptotic variance of the number of real zeros, showing it grows as $ (2K + o(1))\sqrt{n} $ with an explicit constant $ K \approx 0.18198 $.
• To provide a framework that handles test functions with discontinuities and Hölder continuity, enabling broader applicability to linear statistics of zeros.

Proposed method

• Derive new estimates for the two-point correlation function of real zeros of Weyl polynomials using the Kac-Rice formula and conditional Gaussian processes.
• Apply a comparison argument that leverages local laws and repulsion properties of real roots to control fluctuations in the number of zeros.
• Use the Kac-Rice formula to compute the intensity and correlation density of real zeros, expressing them in terms of the covariance structure of the underlying Gaussian process.
• Analyze the conditional distribution of derivatives at zero and at a point $ t $, given that the polynomial vanishes at both, to derive the joint density of zero pairs.
• Employ integral identities involving $ u^{-3/2} $ and $ v^{-3/2} $ to evaluate the expected product of absolute values of derivatives at two points, crucial for computing the correlation function.
• Use a change of variables and special functions (e.g., arcsin, arctan) to evaluate the resulting integrals and derive a closed-form expression for the two-point correlation density $ \rho(0,t) $.

Experimental results

Research questions

• RQ1 Does the number of real zeros of Weyl polynomials satisfy a central limit theorem?
• RQ2 What is the asymptotic variance of the number of real zeros in the Weyl polynomial model?
• RQ3 Can the CLT be established for general linear statistics of real zeros, such as sums over a test function $ h $, rather than just the total count?
• RQ4 How do the correlation functions of real zeros in the Weyl model behave, and what role do local laws and repulsion play in controlling fluctuations?
• RQ5 Can the method used for Weyl polynomials be extended to other classes of random polynomials with non-uniform coefficient scaling?

Key findings

• The number of real zeros $ N_n $ of the Weyl polynomial satisfies a central limit theorem: $ \frac{N_n - \mathbb{E}[N_n]}{(\text{Var}[N_n])^{1/2}} \to N(0,1) $ as $ n \to \infty $.
• The asymptotic variance of $ N_n $ is $ (2K + o(1))\sqrt{n} $, where $ K \approx 0.18198 $ is an explicit constant derived from the correlation structure.
• The expected number of real zeros is $ \mathbb{E}[N_n] = \left(\frac{2}{\pi} + o(1)\right)\sqrt{n} $, consistent with known results for Weyl polynomials.
• The CLT holds not only for the total count but also for general linear statistics $ N_n = \sum_{x \in Z_n} h(x/R_n) $, where $ h $ is bounded, compactly supported, and Hölder continuous away from finitely many points.
• The variance of such linear statistics satisfies $ \text{Var}[N_n] \sim K R_n \|h\|_2^2 $, with the same constant $ K \approx 0.18198 $, confirming universality in the scaling.
• The two-point correlation function $ \rho(0,t) $ is computed explicitly using the Kac-Rice formula and Gaussian process conditioning, yielding a closed-form expression involving $ \arcsin $ and square roots.