Equivalence of Palm measures for determinantal point processes governed by Bergman kernels

TL;DR

This paper proves the mutual absolute continuity of Palm measures for determinantal point processes based on Bergman kernels, under certain domain conditions, and demonstrates their quasi-invariance under diffeomorphisms.

Contribution

It establishes the equivalence of Palm measures for Bergman kernel-based determinantal processes in all dimensions, using the $H^ extinfty(U)$-module structure.

Findings

Mutual absolute continuity of Palm measures is proven.

Results hold for all dimensions of the domain.

Determinantal processes are quasi-invariant under compactly supported diffeomorphisms.

Abstract

For a determinantal point process induced by the reproducing kernel of the weighted Bergman space $A^2(U, \omega)$ over a domain $U \subset \mathbb{C}^d$, we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain $U$ contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the $H^\infty(U)$-module structure of $A^2(U, \omega)$. A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of $U$.