Asymptotic Mass Distribution of Random Holomorphic Sections

TL;DR

This paper proves a central limit theorem for the mass distribution of random holomorphic sections on compact Kähler manifolds and demonstrates quantum ergodicity for almost all such sequences.

Contribution

It establishes a central limit theorem and quantum ergodicity results for random holomorphic sections with $ ext{C}^3$ Hermitian metrics on compact Kähler manifolds.

Findings

Proves a central limit theorem for mass distribution.

Shows quantum ergodicity for almost every sequence.

Applicable to sequences of random holomorphic sections.

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 K\"{a}hler manifold. In addition, we show that almost every sequence of such random holomorphic sections exhibits quantum ergodicity in the sense of Zelditch.