Geometry & Topology

Central limit theorem for spectral partial Bergman kernels

Steve Zelditch and Peng Zhou

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Partial Bergman kernels Πk,E are kernels of orthogonal projections onto subspaces SkH0(M,Lk) of holomorphic sections of the k th power of an ample line bundle over a Kähler manifold (M,ω). The subspaces of this article are spectral subspaces {ĤkE} of the Toeplitz quantization Ĥk of a smooth Hamiltonian H:M. It is shown that the relative partial density of states satisfies Πk,E(z)Πk(z)1A where A={H<E}. Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface A; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values 1 and 0 of 1A. Such “Erf” asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and the central limit theorem.

Article information

Geom. Topol., Volume 23, Number 4 (2019), 1961-2004.

Received: 30 August 2017
Revised: 17 April 2018
Accepted: 30 September 2018
First available in Project Euclid: 16 July 2019

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Mathematical Reviews number (MathSciNet)

Primary: 32A60: Zero sets of holomorphic functions 32L10: Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30] 81Q50: Quantum chaos [See also 37Dxx]

Toeplitz operator partial Bergman kernel interface asymptotics


Zelditch, Steve; Zhou, Peng. Central limit theorem for spectral partial Bergman kernels. Geom. Topol. 23 (2019), no. 4, 1961--2004. doi:10.2140/gt.2019.23.1961.

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