Central Limit theorem for spectral Partial Bergman kernels

Abstract

Partial Bergman kernels ${\Pi }_{k,E}$ are kernels of orthogonal projections onto subspaces $k\subset H^0(M,L^k)$ of holomorphic sections of the $k$th power of an ample line bundle over a Kahler manifold $(M,\omega )$. The subspaces of this article are spectral subspaces $\{{\hat{H}}_k\le E\}$ of the Toeplitz quantization ${\hat{H}}_k$ of a smooth Hamiltonian $H:M\to \mathbb{R}$. It is shown that the relative partial density of states $\frac{{\Pi }_{k,E}(z)}{{\Pi }_k(z)}\to 1_{\mathcal{A}}$ where $\mathcal{A}=\{H<E\}$. 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 $1,0$ 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 central limit theorem

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