Determinantal point processes on complex manifolds: Construction and limit theorems

Abstract

We develop a coordinate-free probabilistic framework for determinantal point processes associated with Bergman kernels on compact complex manifolds. The basic issue is that Bergman kernels are naturally line-bundle-valued: $B_k(x,y)\in \mathrm{Hom}(L_y^k,L_x^k)$. Hence the usual determinantal formula for correlation functions is not literally a scalar determinant unless one first gives it an intrinsic meaning. We rigorously define this determinant and prove that every finite-dimensional Hilbert space of sections of a Hermitian line bundle gives rise to a genuine finite-rank projection determinantal point process on the base manifold. We then isolate a collection of finite-dimensional transfer principles showing how diagonal asymptotics, near-diagonal asymptotics, Schur complements, Toeplitz trace expansions and determinant asymptotics are converted into probabilistic statements. Specializing to $H^0(M,L^k)$, this gives the Bergman ensemble as the geometric analogue of an orthogonal polynomial ensemble, and some of the transfer principles allow us to recover previously known results of Berman.

Complete restructuration of the content around two principles: a coordinate-free construction of the line-bundle-valued kernel DPPs and a series of transfer principles that convert asymptotics of the kernel into limit theorems. 34 pages