[Paper Review] Random complex zeroes, I. Asymptotic normality

Mikhail Sodin, Boris Tsirelson|ArXiv.org|Oct 6, 2002

Geometry and complex manifolds16 references3 citations

TL;DR

This paper establishes asymptotic normality for linear statistics of zeroes of Gaussian random analytic functions in three invariant models: elliptic, flat, and hyperbolic. By analyzing the variance of smooth functionals of the zero set via the covariance structure of the underlying Gaussian process, it proves that normalized linear statistics converge to a normal distribution as the system size $ L \to \infty $, with variance decaying as $ \frac{\kappa}{L} \|\Delta^*h\|^2_{L^2} $, where $ \kappa $ is a universal constant derived from the complex Gaussian chaos expansion.

ABSTRACT

We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of zeroes.

Motivation & Objective

• To establish the asymptotic normality of smooth linear statistics of the zero set of chaotic analytic zero points (CAZP) in three invariant models: elliptic, flat, and hyperbolic.
• To analyze the fluctuations of these linear statistics as the system parameter $ L \to \infty $, which controls the mean number of zeroes per unit area.
• To derive the asymptotic variance of the linear statistics $ Z_L(h) = \sum_{\psi_L(z)=0} h(z) $ in terms of the $ L^2 $-norm of the invariant Laplacian $ \Delta^*h $.
• To demonstrate that the normalized linear statistics converge in distribution to a normal random variable, confirming central limit behavior for smooth functionals of the zero set.

Proposed method

• The authors define three models of Gaussian random analytic functions—elliptic, flat, and hyperbolic—based on unitary invariance and invariant metrics on $ \mathbb{C} \cup \{\infty\} $, $ \mathbb{C} $, and $ \mathbb{D} $, respectively.
• They normalize the process $ \psi_L(z) $ to $ w_L(z) = \psi_L(z)/\|\psi_L(z)\| $, ensuring $ w_L(z) \sim \mathcal{N}_{\mathbb{C}}(0,1) $ pointwise, and define the normalized linear statistics $ Z_L(h) - \mathbb{E}Z_L(h) = \frac{1}{2\pi} \int_{\mathcal{M}} \log|w_L(z)| \Delta^*h(z) \, dm^* $.
• The key step involves computing the variance of $ Z_L(h) $ using the covariance structure of the Gaussian process $ w_L(z) $, specifically the two-point correlation function $ \rho_L(z_1,z_2) = \mathbb{E}[w_L(z_1)\overline{w_L(z_2)}] $, which decays as $ L \to \infty $.
• They show that $ \frac{L}{2\pi} \rho_L(z,0) $ converges weakly to the Dirac delta at 0, implying short-range correlations in the limit.
• Using the chaos expansion of $ \log|\zeta| $ for $ \zeta \sim \mathcal{N}_{\mathbb{C}}(0,1) $, they express the variance as a sum over Hermite coefficients $ c_{2\alpha} $, leading to the asymptotic variance $ \mathbb{E}[Z_L(h)^2] \sim \frac{\kappa}{L} \|\Delta^*h\|^2_{L^2} $, where $ \kappa = \frac{1}{4\pi} \sum_{\alpha \geq 1} \frac{c_{2\alpha}^2}{\alpha} $.
• The proof relies on verifying conditions (2.3) and (2.4) of a general CLT for U-statistics, using the decay of $ |\rho_L(z_1,z_2)|^\beta $ and the weak convergence of $ \frac{L\beta}{2\pi} \rho_L(z,0)^\beta $ to the Dirac measure at 0.

Experimental results

Research questions

• RQ1 Does the linear statistics of the zero set of chaotic analytic zero points converge to a normal distribution as $ L \to \infty $?
• RQ2 What is the asymptotic variance of the linear statistics $ Z_L(h) $, and how does it depend on the test function $ h $?
• RQ3 How does the correlation structure of the underlying Gaussian analytic function $ \psi_L(z) $ affect the fluctuation behavior of the zero set?
• RQ4 Can the central limit theorem for linear statistics be established in the three invariant models (elliptic, flat, hyperbolic) despite differing geometries?
• RQ5 Is the limiting variance universal in form, depending only on the $ L^2 $-norm of the invariant Laplacian $ \Delta^*h $?

Key findings

• The linear statistics $ Z_L(h) $ of the zero set of chaotic analytic zero points converge in distribution to a normal random variable as $ L \to \infty $, establishing asymptotic normality.
• The asymptotic variance of $ Z_L(h) $ is $ \frac{\kappa}{L} \|\Delta^*h\|^2_{L^2(m^*)} $, where $ \kappa = \frac{1}{4\pi} \sum_{\alpha \geq 1} \frac{c_{2\alpha}^2}{\alpha} $ is a universal constant derived from the complex Gaussian chaos expansion.
• The variance decays as $ \frac{1}{L} $, indicating that fluctuations shrink with system size, consistent with central limit behavior.
• The convergence of $ \frac{L}{2\pi} \rho_L(z,0) $ to the Dirac delta at 0 implies that the correlation length of the zero set shrinks as $ L \to \infty $, supporting the CLT.
• The result holds uniformly across the three invariant models (elliptic, flat, hyperbolic), with the same asymptotic variance form, due to the invariance of the normalized process $ w_L(z) $.
• The key technical step is the verification of conditions (2.3) and (2.4) in the CLT framework, relying on the weak convergence of $ \frac{L\beta}{2\pi} |\rho_L(z_1,z_2)|^\beta $ to the Dirac measure at $ z_2 $.