Central limit theorem for spectral partial Bergman kernels

Abstract

Partial Bergman kernels ${\mathrm{\Pi }}_{k,E}$ are kernels of orthogonal projections onto subspaces ${\mathcal{S}}_k\subset H^0\left(M,L^k\right)$ of holomorphic sections of the $k^{\text{ th}}$ power of an ample line bundle over a Kähler manifold $\left(M,\omega \right)$. The subspaces of this article are spectral subspaces $\left\{{Ĥ}_k\le E\right\}$ 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 ${\mathrm{\Pi }}_{k,E}\left(z\right)∕{\mathrm{\Pi }}_k\left(z\right)\to 1_{\mathcal{A}}$ where $\mathcal{A}=\left\{H<E\right\}$. Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface $\partial \mathcal{A}$; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values $1$ and $0$ of $1_{\mathcal{A}}$. 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.