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Central limit theorem for spectral partial Bergman kernels

Steve Zelditch and Peng Zhou

Geometry & Topology 23 (2019) 1961–2004

Partial Bergman kernels Πk,E are kernels of orthogonal projections onto subspaces Sk H0(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 {Ĥk E} 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.

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Toeplitz operator, partial Bergman kernel, interface asymptotics
Mathematical Subject Classification 2010
Primary: 32A60, 32L10, 81Q50
Received: 30 August 2017
Revised: 17 April 2018
Accepted: 30 September 2018
Published: 17 June 2019
Proposed: Gang Tian
Seconded: Leonid Polterovich, Simon Donaldson
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