Central limit theorem for spectral partial Bergman kernels

Zelditch, Steve; Zhou, Peng

doi:10.2140/gt.2019.23.1961

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Central limit theorem for spectral partial Bergman kernels

Steve Zelditch and Peng Zhou

Geometry & Topology 23 (2019) 1961–2004

DOI: 10.2140/gt.2019.23.1961

Abstract

Partial Bergman kernels $Π_{k, E}$ are kernels of orthogonal projections onto subspaces $S_{k} \subset H^{0} (M, L^{k})$ 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 ${Ĥ_{k} \leq E}$ of the Toeplitz quantization $Ĥ_{k}$ of a smooth Hamiltonian $H : M \to ℝ$ . It is shown that the relative partial density of states satisfies $Π_{k, E} (z) ∕ Π_{k} (z) \to 1_{A}$ where $A = {H < E}$ . Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface $\partial A$ ; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values $1$ and $0$ of $1_{A}$ . 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.

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Keywords

Toeplitz operator, partial Bergman kernel, interface asymptotics

Mathematical Subject Classification 2010

Primary: 32A60, 32L10, 81Q50

References

Publication

Received: 30 August 2017

Revised: 17 April 2018

Accepted: 30 September 2018

Published: 17 June 2019

Proposed: Gang Tian

Seconded: Leonid Polterovich, Simon Donaldson

Authors

Steve Zelditch
Department of Mathematics Northwestern University Evanston, IL United States

Peng Zhou
Department of Mathematics Northwestern University Evanston, IL United States	Institut des Hautes Études Scientifiques Bures-sur-Yvette France

Volume 23, issue 4 (2019)