Partition functions of determinantal point processes on polarized Kähler manifolds
Kiyoon Eum
TL;DR
The paper develops a full, two-pronged asymptotic analysis of determinantal point processes on polarized Kähler manifolds, showing that each term in the partition-function expansion is a geometric functional S_j of the Kähler metric that satisfies a cocycle identity. The first three functionals connect to key Kähler-geometric objects: S_0 is the Aubin–Yau functional, S_1 is Mabuchi’s functional, and S_2 extends Liouville-type action to higher dimensions, with Futaki-type invariants obstructing their critical points. The authors provide both Bergman-kernel (TYZ) and Quillen–Ray–Singer routes to the expansion, and they demonstrate a nonpolarized validity for the first three coefficients, while establishing a nonperturbative, path-integral viewpoint and linking the leading higher-dimensional quantum Hall effective action to a (2n+1)-dimensional Chern–Simons form. Overall, the work unifies complex-geometry functionals with probabilistic DPPs and quantum-field-theoretic actions, and confirms a conjecture of Klevtsov on partition-function asymptotics in all dimensions.
Abstract
In this paper, we study the full asymptotic expansion of the partition functions of determinantal point processes defined on a polarized Kähler manifold. We show that the coefficients of the expansion are given by geometric functionals on Kähler metrics satisfying the cocycle identity, whose first variations can be expressed through the TYZ expansion coefficients of the Bergman kernel. In particular, these functionals naturally generalize the Mabuchi functional in Kähler geometry and the Liouville functional on Riemann surfaces. We further show that Futaki-type holomorphic invariants obstruct the existence of critical points of these geometric functionals, extending Lu's formula. We also verify that certain formulas remain valid up to the third coefficient without assuming polarization. Finally, we discuss the relation of our results to the quantum Hall effect (QHE), where the determinantal point process provides a microscopic model. In particular, we recover the higher-dimensional effective Chern-Simons actions derived in the physics literature and confirm a conjecture of Klevtsov on the form of the partition function asymptotics.