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Central Limit Theorem for Intersection Currents of Gaussian Holomorphic Sections

Bin Guo

Abstract

In 2010, Shiffman and Zelditch proved a central limit theorem (CLT) for smooth statistics of Gaussian random zeros in codimension one over compact Kähler manifolds. They raised the question of whether this result admits a two-fold generalization -- to arbitrary codimensions and to both smooth and numerical statistics -- which has remained open since then. In this paper we resolve this long-standing problem. We establish a universal CLT that holds for both types of statistics arising from several independent Gaussian sections, thereby fully extending the Shiffman--Zelditch theorem. The proof builds on a new geometric framework that lifts the probabilistic tools of Wiener chaos and Feynman diagrams from scalar processes to random currents on complex manifolds, providing a robust mechanism for analyzing fluctuations in random complex geometry beyond the classical codimension-one setting.

Central Limit Theorem for Intersection Currents of Gaussian Holomorphic Sections

Abstract

In 2010, Shiffman and Zelditch proved a central limit theorem (CLT) for smooth statistics of Gaussian random zeros in codimension one over compact Kähler manifolds. They raised the question of whether this result admits a two-fold generalization -- to arbitrary codimensions and to both smooth and numerical statistics -- which has remained open since then. In this paper we resolve this long-standing problem. We establish a universal CLT that holds for both types of statistics arising from several independent Gaussian sections, thereby fully extending the Shiffman--Zelditch theorem. The proof builds on a new geometric framework that lifts the probabilistic tools of Wiener chaos and Feynman diagrams from scalar processes to random currents on complex manifolds, providing a robust mechanism for analyzing fluctuations in random complex geometry beyond the classical codimension-one setting.
Paper Structure (48 sections, 30 theorems, 392 equations, 10 figures, 1 table)

This paper contains 48 sections, 30 theorems, 392 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Expectation and variance of random zeros
  3. Review of the codimension-one CLT
  4. Our contribution: a geometric chaos framework
  5. Organization of the paper
  6. Proof of the Main Theorem assuming Lemmas \ref{['lem:truncated-case']} and \ref{['lem:reduce-process']}
  7. Chaos currents $\mathcal{C}^N_\alpha$
  8. Proof of the main theorem
  9. Correlations of chaos currents
  10. Feynman diagrams and the Wick formula
  11. Intrinsic interpretation
  12. Efficient encoding of edge information: associated directed multigraphs $\gamma^*$
  13. Feynman-correlation currents
  14. $p$-correlation of chaos currents
  15. $2$-correlation of (truncated) integral currents
  16. ...and 33 more sections

Key Result

Theorem 1.2

Let $(L,h) \to (M,\omega)$ with $\omega = \pi c_1(L,h)$. Endow $H^0(M, L^N)$ with the standard Gaussian measure. Let $\varphi$ be a real-valued $(m-1,m-1)$-form with $\mathscr{C}^3$ coefficients such that $\partial\bar{\partial}\varphi \neq 0$, and let $s^N \in H^0(M,L^N)$ be a random section. Then

Figures (10)

  • Figure 1: The width of each distribution represents the standard deviation $\sigma=\sqrt{\operatorname{Var}}$. Expectations (dashed lines) converge to macroscopic equidistribution limits (down arrows).
  • Figure 2: Example of $\gamma\in\Gamma(3,2,2,1)$
  • Figure 3: the corresponding $\gamma^*$
  • Figure 4: $\gamma_1\in\Gamma(1,1,1,1,1,1)$
  • Figure 5: $\gamma_2\in\Gamma(1,1,1,0,1,1)$
  • ...and 5 more figures

Theorems & Definitions (67)

  • Remark 1.1
  • Theorem 1.2: MR2742043
  • Definition 2.1: Chaos currents
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['lem:key-reduce']}
  • proof : Proof of the Main Theorem
  • Definition 3.1: Feynman diagram
  • Definition 3.2: Value of a labelled diagram
  • ...and 57 more