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Asymptotic Mass Distribution of Random Holomorphic Sections

Turgay Bayraktar, Afrim Bojnik

TL;DR

This work studies the asymptotic behavior of mass distributions for random holomorphic sections over a compact Kähler manifold under a Diophantine curvature condition on a sequence of line bundles. It proves a central limit theorem for linear statistics of mass and establishes almost-sure quantum ergodicity (mass equidistribution) by leveraging refined Bergman kernel asymptotics (diagonal growth, off-diagonal decay, near-diagonal expansions). The analysis blends complex-analytic tools with probabilistic limit theorems, notably the Sodin–Tsirelson nonlinear CLT for Gaussian processes, to obtain universal Gaussian fluctuations of mass measures. By extending results beyond tensor powers to general line-bundle sequences, the paper provides a robust framework for mass distribution and quantum ergodicity in geometric quantization.

Abstract

In this note, we prove a central limit theorem for the mass distribution of random holomorphic sections associated with a sequence of positive line bundles endowed with $\mathscr{C}^3$ Hermitian metrics over a compact Kähler manifold. In addition, we show that almost every sequence of such random holomorphic sections exhibits quantum ergodicity in the sense of Zelditch.

Asymptotic Mass Distribution of Random Holomorphic Sections

TL;DR

This work studies the asymptotic behavior of mass distributions for random holomorphic sections over a compact Kähler manifold under a Diophantine curvature condition on a sequence of line bundles. It proves a central limit theorem for linear statistics of mass and establishes almost-sure quantum ergodicity (mass equidistribution) by leveraging refined Bergman kernel asymptotics (diagonal growth, off-diagonal decay, near-diagonal expansions). The analysis blends complex-analytic tools with probabilistic limit theorems, notably the Sodin–Tsirelson nonlinear CLT for Gaussian processes, to obtain universal Gaussian fluctuations of mass measures. By extending results beyond tensor powers to general line-bundle sequences, the paper provides a robust framework for mass distribution and quantum ergodicity in geometric quantization.

Abstract

In this note, we prove a central limit theorem for the mass distribution of random holomorphic sections associated with a sequence of positive line bundles endowed with Hermitian metrics over a compact Kähler manifold. In addition, we show that almost every sequence of such random holomorphic sections exhibits quantum ergodicity in the sense of Zelditch.
Paper Structure (7 sections, 12 theorems, 111 equations)

This paper contains 7 sections, 12 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reference Covers and Special Metric Structures
  4. Bergman Kernel Asymptotics
  5. Mass Equidistribution for Random Holomorphic Sections
  6. Asymptotic Normality for Masses of Random Sections
  7. Central Limit Theorem for Nonlinear Functionals of Complex Gaussian Processes

Key Result

Theorem 1.1

Let $\{(L_p, h_p)\}_{p \geq 1}$ be a sequence of positive holomorphic line bundles over a compact Kähler manifold $(X, \omega)$ of complex dimension $n$, satisfying the Diophantine condition (appr), and endowed with Hermitian metrics of class $\mathscr{C}^3$ such that $\|h_p\|_3^{1/3} / \sqrt{A_p} \ converge in distribution to the standard real Gaussian law $\mathcal{N}_{\mathbb{R}}(0,1)$ as $p \t

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2: Reference Cover Lemma
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 11 more