Central Limit theorem for spectral Partial Bergman kernels

TL;DR

This paper establishes a central limit theorem for spectral partial Bergman kernels on Kahler manifolds, showing Gaussian error function asymptotics at spectral edges, revealing universal edge effects similar to classical probability laws.

Contribution

It introduces a CLT for spectral partial Bergman kernels and describes their universal Gaussian error function asymptotics at spectral boundaries.

Findings

Relative partial density converges to indicator function of sublevel set

Density exhibits Gaussian error function asymptotics at the interface

Asymptotics resemble law of large numbers and CLT

Abstract

Partial Bergman kernels $\Pi_{k, E}$ are kernels of orthogonal projections onto subspaces $\mathcal{k} \subset H^0(M, L^k)$ 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 $\{\hat{H}_k \leq E\}$ 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