Toeplitz operators and zeros of square-integrable random holomorphic sections

2024/04/24 by Alexander Drewitz, Drewitz, Alexander, Bingxiao Liu +3
Mathematics · #30C15 #32A60 #47B35 #53D50 #60D05 #Advanced Algebra and Geometry #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Holomorphic and Operator Theory #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.2404.15983

openalex publication_date 2024/04/24 · openalex created_date 2024/04/26 · openalex updated_date 2026/07/28

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.

Related

• Berezin-Toeplitz operators, Kodaira maps, and random sections
• Gaussian holomorphic sections on noncompact complex manifolds
• Random Systems of Holomorphic Sections of a Sequence of Line bundles on Compact Kähler Manifolds
• Asymptotic expansion of the variance of random zeros on complex manifolds
• Asymptotic Mass Distribution of Random Holomorphic Sections