Bergman Determinantal Point Process

• Bergman DPP is a determinantal point process defined via Bergman kernels arising from orthogonal projections in L²-spaces of holomorphic functions.
• Its construction utilizes reproducing kernel Hilbert space theory to model correlations and asymptotic behaviors in diverse geometric settings.
• The framework applies to both infinite- and finite-rank ensembles on domains like the unit disk and compact complex manifolds, aiding numerical integration and simulation.

A Bergman determinantal point process is a determinantal point process whose correlation kernel is a Bergman kernel, or more generally a Bergman projection kernel, attached to a Hilbert space of holomorphic functions or holomorphic sections. The term covers several distinct but structurally parallel models: infinite-rank processes on the unit disk and on weighted Bergman spaces over bounded pseudoconvex domains, finite-rank “Bergman ensembles” on compact complex manifolds obtained from $H^0(X,L^k)$, and hyperbolic or complex-hyperbolic variants governed by invariant Bergman kernels (Lemoine et al., 2024, Bufetov, 2021, Eum, 2024). In all cases the core mechanism is the same: an orthogonal projection in an $L^2$-space produces a reproducing kernel, and that kernel defines the determinantal correlations.

1. Projection-kernel definition

Let $(E,\lambda)$ be a measure space, and let $K$ be a kernel such that the $n$-point correlation functions satisfy

$$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$

A point process with this property is determinantal. When the associated integral operator is Hermitian, locally trace class, and satisfies $0\le K\le I$, the Macchi–Soshnikov / Shirai–Takahashi theorem yields existence and uniqueness of the corresponding DPP; in particular, any locally trace class orthogonal projection defines a DPP (Lazag, 24 Mar 2025).

Bergman DPPs are projection DPPs. If $H\subset L^2(E,\lambda)$ is a reproducing kernel Hilbert space of holomorphic functions, and $P_H$ is the orthogonal projection onto $H$, then the integral kernel of $L^2$0 is the Bergman kernel. The reproducing property has the form

$L^2$1

and the kernel is Hermitian and idempotent at the operator level. This is the basic construction on the unit disk, on weighted Bergman spaces over domains in $L^2$2, and on spaces of holomorphic sections of positive line bundles (Bufetov, 2021, Eum, 2024).

In the compact-manifold setting, the process is finite. If $L^2$3 is an orthonormal basis of $L^2$4, the Bergman kernel is

$L^2$5

and the associated Bergman ensemble has joint density

$L^2$6

with $L^2$7. Because the kernel is the orthogonal projector onto a finite-dimensional subspace, the process has exactly $L^2$8 points (Lemoine et al., 2024).

2. Principal geometric realizations

The phrase “Bergman determinantal point process” does not denote a single model. It denotes a family of determinantal processes attached to Bergman-type spaces in several geometries (Bufetov et al., 2017, Lemoine, 2022).

Setting	Hilbert space / kernel	Process type
Unit disk $L^2$9	$(E,\lambda)$0, $(E,\lambda)$1 with $(E,\lambda)$2, or $(E,\lambda)$3 with Lebesgue measure	Infinite-rank
Weighted disk	$(E,\lambda)$4, $(E,\lambda)$5	Infinite-rank
Bounded pseudoconvex $(E,\lambda)$6	$(E,\lambda)$7, kernel $(E,\lambda)$8	Infinite-rank
Compact complex manifold $(E,\lambda)$9	$K$0, kernel $K$1	Finite-rank, exactly $K$2 points
Hyperbolic disk / complex hyperbolic space	Weighted Bergman kernels and Bergman projection kernels	Infinite-rank

On the unit disk, the standard Bergman space $K$3 with normalized Lebesgue measure $K$4 has kernel

$K$5

whereas with Lebesgue measure as background measure the kernel is

$K$6

This normalization dependence is standard. The corresponding process is also the zero set of the Gaussian analytic function

$K$7

with i.i.d. standard complex Gaussian coefficients (Lin et al., 2024, Bufetov, 2021).

Weighted disk models include the classical weighted Bergman spaces $K$8, $K$9, with kernel

$n$0

More general generalized Bergman spaces on $n$1 also define DPPs under the integrability condition

$n$2

(Bufetov et al., 2014).

On bounded pseudoconvex domains $n$3, one considers

$n$4

where $n$5. Its reproducing kernel $n$6 defines a DPP $n$7 with respect to $n$8. This is an infinite-rank Bergman DPP with a semiclassical parameter $n$9 (Eum, 2024).

On compact complex manifolds, Bergman DPPs arise from a positive holomorphic line bundle $$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$0 and its tensor powers $$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$1. The Bergman kernel of $$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$2 yields a finite determinantal ensemble with exactly $$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$3 points (Lemoine, 2022).

Complex-hyperbolic and hyperbolic models include the Bergman DPP on $$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$4 induced by the kernel

$$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$5

as well as weighted Bergman kernels on the Poincaré disk such as

$$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$6

with reference measure $$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$7 (Bufetov et al., 2021, Demni et al., 2017).

3. Asymptotic regimes and universal limits

For Bergman ensembles on compact complex manifolds, the principal asymptotic regime is $$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$8. In local normal coordinates at a point $$\rho_n(x_1,\dots,x_n)=\det\big(K(x_i,x_j)\big)_{1\le i,j\le n}.$$9, the Bergman kernel admits a near-diagonal expansion whose leading term is a Gaussian kernel on $0\le K\le I$0. More precisely, after rescaling by $0\le K\le I$1 and $0\le K\le I$2,

$0\le K\le I$3

where

$0\le K\le I$4

and $0\le K\le I$5 are the eigenvalues of the curvature matrix at the base point. The rescaled $0\le K\le I$6-point correlation functions converge to

$0\le K\le I$7

so the local limit is a multidimensional generalization of the infinite Ginibre ensemble (Lemoine, 2022).

The same compact-manifold framework yields global laws. If

$0\le K\le I$8

then $0\le K\le I$9 converges in probability, in the weak topology, to the equilibrium measure

$H\subset L^2(E,\lambda)$0

For weighted variants of the process, the empirical measures satisfy a large deviation principle with speed $H\subset L^2(E,\lambda)$1, and the rate function is the Legendre–Fenchel transform of the Mabuchi functional increment

$H\subset L^2(E,\lambda)$2

(Lemoine, 2022).

A distinct asymptotic regime appears on bounded pseudoconvex domains. There the emphasis is not fixed-cardinality ensembles but the large-$H\subset L^2(E,\lambda)$3 behavior of the infinite-rank weighted Bergman kernels $H\subset L^2(E,\lambda)$4. For $H\subset L^2(E,\lambda)$5-admissible $H\subset L^2(E,\lambda)$6, the scaled cumulant generating function satisfies

$H\subset L^2(E,\lambda)$7

where

$H\subset L^2(E,\lambda)$8

This connects weighted Bergman DPPs on bounded domains to pluripotential energy functionals (Eum, 2024).

At the flat-model level, the Heisenberg family on $H\subset L^2(E,\lambda)$9 has kernel

$P_H$0

with respect to Gaussian background measure. In the operator-theoretic framework of partial isometries and reproducing kernels, this is the Bargmann–Fock model that underlies Ginibre-type limits (Katori et al., 2019).

4. Hyperbolic Bergman processes and fluctuation theory

On the Poincaré disk, the lowest hyperbolic Landau level $P_H$1 gives the weighted Bergman kernel

$P_H$2

and the associated hyperbolic-type point process $P_H$3 is a Bergman DPP. More generally, higher hyperbolic Landau levels yield kernels $P_H$4 involving Jacobi polynomials and define generalized Bergman-type DPPs (Demni et al., 2017).

For the number $P_H$5 of particles inside the Euclidean disk $P_H$6, the hyperbolic-type process satisfies

$P_H$7

In the Bergman case $P_H$8, the kernel expansion leads to a full distributional description: $P_H$9 has the same distribution as a sum of independent Bernoulli random variables with explicit success probabilities

$H$0

For $H$1, this recovers the Peres–Virág formula

$H$2

(Demni et al., 2017).

On the upper half-plane, the affine ensemble associated with the $H$3 group contains the weighted Bergman kernel as the ground state: $H$4 For hyperbolic disks $H$5, the variance has the exact form

$H$6

and asymptotically

$H$7

The weighted Bergman DPP arises as the case $H$8, equivalently the lowest hyperbolic Landau level (Abreu et al., 2022).

A more recent structural theorem studies homogeneous projection DPPs on non-elementary Gromov hyperbolic spaces and applies explicitly to Bergman projection kernels on standard hyperbolic spaces. If $H$9 is a homogeneous DPP on a homogeneous hyperbolic space $L^2$00, then there exists $L^2$01 such that

$L^2$02

Since projection DPPs also satisfy $L^2$03, the variance is of the same order as the expectation, and such processes are never hyperuniform (Lazag, 24 Mar 2025).

A potential source of confusion is that another hyperbolic line of work interprets the asymptotic $L^2$04 for affine ensembles as growth like boundary length (Abreu et al., 2022), whereas the general homogeneous theorem concludes non-hyperuniformity (Lazag, 24 Mar 2025). This suggests that fluctuation terminology in hyperbolic geometry is sensitive to how window size is parameterized and compared with ambient volume growth.

5. Palm measures, conditional laws, and quasi-invariance

For generalized Bergman DPPs on the unit disk, a central structural result is the equivalence of reduced Palm measures. Under the condition

$L^2$05

all reduced Palm measures of all orders are mutually equivalent, and their Radon–Nikodym derivatives are expressed by regularized multiplicative functionals built from finite Blaschke products. If

$L^2$06

then the derivative of the Palm measure at $L^2$07 has the form

$L^2$08

with $L^2$09 a regularized additive functional. In this Bergman setting the process is explicitly described as non-rigid, in contrast with the generalized Fock case on $L^2$10 (Bufetov et al., 2014).

The higher-dimensional analogue holds on domains $L^2$11 for weighted Bergman spaces $L^2$12. If $L^2$13 contains a non-constant bounded holomorphic function, then the determinantal measure $L^2$14 is equivalent to its arbitrary reduced Palm measure of any order. A corollary is quasi-invariance under compactly supported diffeomorphisms of $L^2$15 (Bufetov et al., 2017).

On the unit disk with the classical Bergman kernel

$L^2$16

the conditional law inside a relatively compact region $L^2$17, given the exterior configuration $L^2$18, admits an explicit $L^2$19-ensemble form. The conditional kernel is

$L^2$20

where $L^2$21 is a regularized Blaschke-type multiplicative functional, and the reduced Palm measure satisfies

$L^2$22

This gives a concrete Gibbs-like description of the Bergman DPP conditioned on the outside configuration (Bufetov, 2021).

6. Reconstruction, quadrature, and simulation

Bergman DPPs have been used as numerical-integration nodes on compact complex manifolds. For a holomorphic line bundle $L^2$23 and Bergman ensemble with kernel $L^2$24, the estimator

$L^2$25

is unbiased: $L^2$26 For Lipschitz $L^2$27 supported in the weak bulk,

$L^2$28

and the mean squared error decays as

$L^2$29

This is faster than the $L^2$30 rate of i.i.d. Monte Carlo (Lemoine et al., 2024).

On the unit disk, vector-valued linear statistics in the Bergman space provide another analytical use. For

$L^2$31

the squared norm $L^2$32 satisfies a law of large numbers. As an application, for almost every Bergman DPP configuration the weighted Poincaré series

$L^2$33

cannot converge simultaneously for all $L^2$34 whenever $L^2$35 (Lin et al., 2024).

On complex hyperbolic spaces, Patterson–Sullivan interpolation uses a Bergman DPP configuration as a random discrete set from which pluriharmonic, weighted Bergman, and harmonic Hardy functions can be reconstructed. For super-critical weighted Bergman spaces, simultaneous uniform interpolation holds on the unit ball of the space; for the unweighted Bergman space $L^2$36, uniform simultaneous interpolation is impossible (Bufetov et al., 2021).

Simulation questions motivate another line of work on the unit disk Bergman DPP. Restricting the process to the closed ball $L^2$37, the Mercer decomposition is explicit: $L^2$38 The expected number of restricted points is

$L^2$39

For truncation to $L^2$40 eigenvalues, the Kantorovich–Rubinstein Wasserstein distance satisfies

$L^2$41

and the discrepancy probability obeys the same exponential bound. The same work proves general deviation inequalities for the cardinality of any trace-class DPP (Driot et al., 10 Jul 2025).

These applications separate two recurring roles of Bergman DPPs. One role is geometric and asymptotic: Bergman kernels encode complex geometry, and the process exposes that structure probabilistically. The other is algorithmic and functional-analytic: the projection-kernel form yields tractable linear statistics, concentration estimates, and explicit finite approximations.