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Toeplitz operators and zeros of square-integrable random holomorphic sections

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arxiv 2404.15983 v1 pith:M3KZPSZL submitted 2024-04-24 math.CV math-phmath.MP

Alexander Drewitz , Bingxiao Liu , George Marinescu This is my paper · ORCID
classification math.CVmath-phmath.MP
keywords zerosasymptoticberezin-toeplitzfunctionholomorphiclimitrandomsections
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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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry

    math.CV 2025-02 conditional novelty 8.0 of 10

    Small eigenvalues of Toeplitz operators with proper support decay exponentially, and their logarithmic distribution is governed by the Mabuchi geodesic speed to the Lebesgue envelope.

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    math.CV 2025-06 conditional novelty 7.0 of 10

    For Bernstein-Markov measures on big line bundles, Bergman measures concentrate on the diagonal and Toeplitz operators close under composition, with spectra governed by the equilibrium measure.

  3. Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds

    math.CV 2025-05 conditional novelty 6.0 of 10

    For a compact Levi non-degenerate CR manifold, the semiclassical spectral projector of a Levi-elliptic Toeplitz operator is, modulo a negligible kernel, the sum of two oscillatory integrals with complex phases, with e...

  4. A survey on asymptotic equilibrium distribution of zeros of random holomorphic sections

    math.CV 2025-04 conditional novelty 2.0 of 10

    A survey of quantitative equidistribution of zeros of random holomorphic sections toward equilibrium currents, emphasizing pluripotential theory and Bergman kernels, with two theorem variants proved.

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