Toeplitz operators and zeros of square-integrable random holomorphic sections

TL;DR

This paper develops a probabilistic framework using Wiener spaces to study the asymptotic distribution of zeros of Gaussian holomorphic sections in Berezin-Toeplitz quantization, revealing equidistribution, large deviations, and CLTs.

Contribution

It introduces a novel probabilistic model for Berezin-Toeplitz quantization on complex manifolds, analyzing zero distributions with new asymptotic expansion techniques.

Findings

Zeros of Gaussian sections become equidistributed in the semiclassical limit.

Established large deviation principles for zero distributions.

Proved central limit theorems for fluctuations of zeros.

Abstract

We use the theory of abstract Wiener spaces to construct a probabilistic model for Berezin-Toeplitz quantization on a complete Hermitian complex manifold endowed with a positive line bundle. We associate to a function with compact support (a classical observable) a family of square-integrable Gaussian holomorphic sections. Our focus then is on the asymptotic distributions of their zeros in the semiclassical limit, in particular, we prove equidistribution results, large deviation estimates, and central limit theorems of the random zeros on the support of the given function. One of the key ingredients of our approach is the local asymptotic expansions of Berezin-Toeplitz kernels with non-smooth symbols.