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: $B_k(x,y)\in\operatorname{Hom}(L_y^k,L_x^k)$. 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.