## Geometry & Topology

### Central limit theorem for spectral partial Bergman kernels

#### Abstract

Partial Bergman kernels $Πk,E$ are kernels of orthogonal projections onto subspaces $Sk⊂H0(M,Lk)$ of holomorphic sections of the power of an ample line bundle over a Kähler manifold $(M,ω)$. The subspaces of this article are spectral subspaces ${Ĥk≤E}$ of the Toeplitz quantization $Ĥk$ of a smooth Hamiltonian $H:M→ℝ$. It is shown that the relative partial density of states satisfies $Πk,E(z)∕Πk(z)→1A$ where $A={H. 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

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

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

https://projecteuclid.org/euclid.gt/1563242523

Digital Object Identifier
doi:10.2140/gt.2019.23.1961

Mathematical Reviews number (MathSciNet)
MR3981005

#### Citation

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. https://projecteuclid.org/euclid.gt/1563242523

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