Partial Bergman kernels and determinantal point processes on K\"ahler manifolds

TL;DR

This paper derives detailed asymptotics for partial Bergman kernels on Kähler manifolds with circle actions and uses these results to analyze the statistical behavior of related determinantal point processes, showing convergence to a normal distribution.

Contribution

It provides the first full off-diagonal asymptotics of equivariant and partial Bergman kernels on Kähler manifolds with bounded geometry, and applies these to determine the asymptotic distribution of linear statistics of associated point processes.

Findings

Asymptotics of partial Bergman kernels computed

Distribution of linear statistics converges to normal

Variance involves H^1 and H^{1/2} norms

Abstract

We compute the full off-diagonal asymptotics of the equivariant and partial Bergman kernels associated with a circle action on a prequantized K\"ahler manifold with bounded geometry at infinity, then use these results to compute the asymptotics of the linear statistics of the associated determinantal point process as the number of points grows to infinity, showing that its distribution converges to a centered normal variable with variance given by the sum of an $H^1$-norm squared in the bulk and an $H^{1/2}$-norm squared on the boundary of the associated droplet.