# Supervised Manifold Learning via Random Forest Geometry-Preserving
Proximities*

Jake S. Rhodes  Department of Statistics
Brigham Young University
Provo, Utah, USA
<EMAIL_ADDRESS>
###### Abstract

Manifold learning approaches seek the intrinsic, low-dimensional data
structure within a high-dimensional space. Mainstream manifold learning
algorithms, such as Isomap, UMAP, $t$-SNE, Diffusion Map, and Laplacian
Eigenmaps do not use data labels and are thus considered unsupervised.
Existing supervised extensions of these methods are limited to classification
problems and fall short of uncovering meaningful embeddings due to their
construction using order non-preserving, class-conditional distances. In this
paper, we show the weaknesses of class-conditional manifold learning
quantitatively and visually and propose an alternate choice of kernel for
supervised dimensionality reduction using a data-geometry-preserving variant
of random forest proximities as an initialization for manifold learning
methods. We show that local structure preservation using these proximities is
near universal across manifold learning approaches and global structure is
properly maintained using diffusion-based algorithms.

###### Index Terms:

supervised learning, manifold learning, random forest, data visualization,
data geometry

##  I Introduction

Manifold learning algorithms are often used for exploratory data analysis.
They are typically applied to noisy data in an attempt to find meaningful
patterns or relationships across time, classes, or variables [1]. Most
manifold learning approaches are unsupervised in that they do not use
auxiliary information (e.g., data labels) in the embedding construction
process. In many contexts, only unsupervised models are applicable as
auxiliary information can be expensive or inaccessible. However, when
available, label information can provide valuable insights into the data’s
intrinsic structure relative to the labels. Subjecting the embedding process
to the use of auxiliary information can help to uncover a data geometry
unattainable without labels. In this paper, we discuss weaknesses of current
supervised manifold-learning approaches and show improvements on existing
methods by applying a new variant of a random forest-based [2] similarity
measure in a manifold-learning setting. We use Geometry- and Accuracy-
Preserving proximities (RF-GAP [3]) and demonstrate their ability to
meaningfully encode a similarity measure and subsequent embedding that
naturally incorporates labels. As opposed to distance or similarity measures
which condition upon class labels to artificially exaggerate the separation of
points of opposing classes, random forest proximities serve as a measure of
similarity that uses labels (continuous, categorical, or otherwise) in a
manner consistent with the model’s learning. Additionally, forest-based
proximities appropriately denoise the data, providing a meaningful metric or
graph for the embedding process.

##  II Supervised Manifold Learning

Manifold-learning algorithms use distance or similarity graphs to encode local
data structure. For example, Isomap forms a $k$-nearest neighbor ($k$-NN)
graph using Euclidean distance and seeks the shortest path between
observations to approximate true geodesic distances upon which multi-
dimensional scaling is applied [4]. Diffusion Map (DM) uses a Gaussian kernel
applied to a $k$-NN graph to form local similarities upon which
eigendecomposition is applied [5]. T-distributed Stochastic Neighbor
Embedding, or $t$-SNE, estimates probabilities as a normalized Gaussian kernel
to define similarities between points. The Kullback-Leibler (KL) divergence
between these and a lower-dimensional mapping is estimated via gradient
descent to form the target embedding [6].

Each of these methods is unsupervised; data labels are not used in any part of
the embedding process. However, supervised variants of these methods have been
developed. Most of these supervised extensions of the algorithms adapt the
existing algorithm at the distance- or similarity-learning level. In some
cases, distances are rescaled [7], additively incremented [8], or otherwise
adapted conditionally upon class association [9]. Often, these dissimilarity
measures can provide perfect linear separation where such discrimination is
not possible using traditional classifiers. See Equation 1, for an example of
a class-conditional dissimilarity, where $D(.,.)$ denotes a distance function
(e.g., Euclidean), $\beta$ is usually set as the average distance between
points, $\alpha$ lessens separation between similar points of opposing
classes, and $y_{i}$, $y_{j}$ are the respective labels of $x_{i}$ and
$x_{j}$. This dissimilarity has been used to create supervised variants of
$t$-SNE [10], Isomap [9], Locally-Linear Embedding [11], and Laplacian
Eigenmaps [12]. In each of these extensions, the within-class structure is
partially maintained, but manifold structures are distorted at a global level
as a result of exaggerated class separation. Such dissimilarity measures are
order non-preserving bijections [13].

$D^{\prime}\left(\mathbf{x}_{i},\mathbf{x}_{j}\right)=\left\\{\begin{array}[]{ll}\sqrt{1-e^{\frac{-D^{2}\left(\mathbf{x}_{i},\mathbf{x}_{j}\right)}{\beta}}}&y_{i}=y_{j}\\\
\sqrt{e^{\frac{D^{2}\left(\mathbf{x}_{i},\mathbf{x}_{j}\right)}{\beta}}}-\alpha&y_{i}\neq
y_{j}\end{array}\right.$ (1)

Figure 1: This figure provides UMAP-based 2-dimensional representations of the
cars dataset [14]. The left figure is the original UMAP [15] implementation
which shows distinct clusters of overlapping classes. A supervised version of
UMAP [16] (center) provides perfect class separation as well as distinctive
clusters, which does not coincide with a cross-validated $k$-NN accuracy of
0.899. The RF-GAP-based UMAP implementation (right) shows observational
relationships which correspond to variable interactions within the dataset.

These approaches are problematic in several ways: (1) The class-conditional
distances form an attempt to maintain with-class structure but cause
disruption between classes. Artificial class separation diminishes inter-class
relationships, thus distorting the global data structure. (2) The manifold
disruption reduces the integrity of resulting downstream tasks. For example,
classification tasks following dimensionality reduction can have
unrealistically low error rates. (3) Class-conditional measures do not provide
an avenue for continuously-valued labels. These extensions have not been
adapted to regression problems. (4) These approaches are not extendable to
new, unlabeled points (e.g., a test set used for subsequent predictions). To
overcome each of these weaknesses, we propose the use of random forests [2, 3]
to generate supervised similarities to be used in manifold learning.

##  III Random Forest Proximities

Random forests [2] provide a number of benefits for prediction problems that
are supportive of metric learning. For example, random forests apply to both
regression and classification problems, handle mixed variable types, provide
an unbiased estimate of the generalization error, are insensitive to monotonic
transformations, are relatively robust to outliers, and provide a natural way
of assessing variable importance, ignoring noise variables in the presence of
meaningful ones.

Random forests form an ensemble in binary-recursive decision trees each of
which partitions a bootstrap sample of the training data. Observations within
the bootstrap sample are called in-bag, while those in the training data not
included in the sample are called out-of-bag, or OOB. Each partition forms a
decision space used for classification or regression. The partitions naturally
form a channel for generating similarity measures using data labels. These
similarities are referred to in the literature as random forest proximities.

Leo Breiman originally defined the proximity between two observations,
$x_{i}$, and $x_{j}$, denoted by $p(x_{i},x_{j})$, as the number of terminal
nodes they share across all trees, divided by the number of trees in the
forest [17]. This simple approach applies to all training points regardless of
bootstrap status. As a result, proximities constructed on the training set
tend to slightly overemphasize class segregation. To overcome this, an
alternative formulation was derived to only calculates pair-wise similarities
between points $x_{i}$ and $x_{j}$ using trees in which both of these
observations are OOB [18]. Subsequently, these proximities combat overinflated
class separation. However, it has been shown that OOB-only proximities do not
fully benefit from the random forest’s learning and are a noisier similarity
measure [3].

In [19], the authors demonstrated that random forests behave like a nearest-
neighbor regressor with an adaptive bandwidth. That is, random forest
predictions (in the regression context) can be determined as a weighted sum
using a kernel function, as shown in Equation 2.

$\hat{y}_{i}(k)=\sum_{j\neq i}k(x_{i},x_{j})y_{j}$ (2)

Here, $k$ is a weighted kernel function determined by the number of training
examples sharing a terminal node with $x_{i}$. This is comparable to other
kernel methods, such as the SVM, which uses a kernel to define similarity and
ultimately the decision space. Ideally, random forest proximities should serve
as a kernel capable of mimicking random forest predictions. Using normalized
proximities as weights can serve as a test for the proximities’ consistency
with the forest’s learning. In this regard, existing random forest proximity
formulations do not adequately incorporate the forest’s learning [3].

Both the original formulation [17], as well as that using only OOB
observations [18], are not capable of reconstructing the random forest
predictions for two reasons: (1) Random forests train on a set of bootstrap
(in-bag) samples and predict on another set (the OOB samples or a test set).
The original formulation doesn’t discriminate between in-bag or OOB
observations, and the OOB proximity definition does not use in-bag samples in
their construction. (2) Decision tree voting takes into account the number of
in-bag observations within a given terminal node, while the proximities are
constructed without regard to the number of “voting points” [19]. To construct
proximities that serve as a kernel for random forest prediction, these two
points must be accounted for.

In [3], the authors propose a new proximity formulation capable of
reconstructing random forest OOB and test predictions as a kernel method. They
call these proximities Random Forest-Geometry- and Accuracy-Preserving
proximities (RF-GAP) and show improvement across multiple applications using
this new definition, including data imputation, outlier detection, and
visualization via MDS.

The RF-GAP proximities, however, do not form a proper kernel function as
originally defined. They are asymmetric and self-similarity is defined to be 0
to account for the kernel prediction problem. To overcome this, we normalize
the similarities to set the maximum similarity to 1, symmetrize the
proximities, and define the diagonal entries to be 1. In doing so, the
proximities can serve in any capacity which requires a kernel matrix. Using
this modified RF-GAP formulation, these proximities serve as a similarity
measure that overcomes the weaknesses of the class-conditional supervised
distances in the following ways: (1) Rather than conditionally adapting an
existing distance measure and thereby distorting the global data structure,
random forest proximities provide a measure of local similarity which
partially retains global information through the forest’s recursive splitting
process. Therefore, instead of exaggerating class separation, natural
observational relationships are retained, as can be seen in low-dimensional
visual representations (see Figure 1). (2) Proximities formed using OOB data
points retain the random forest’s learning, thus, downstream task integrity is
not jeopardized but relevant supervised information is retained. (3) Random
forests are not limited to classification problems but also work with
continuous labels. This provides an avenue for supervised metric learning in a
regression context, as shown in Figure 2. (4) A trained random forest model
can extend similarity measures to unlabeled or out-of-sample observations,
providing a means for semi-supervised metric learning or subsequent
prediction. Additionally, noise variables are not likely to be used for
splitting unless relevant variables are not included in the random subset of
splitting variables. Subsequently, generated proximities naturally account for
variable importance and can serve as a means of denoising.

We show that supervised manifold learning methods generally improve with the
use of RF-GAP proximities. We compare the embedding mapping using common
manifold learning algorithms including Isomap, $t$-SNE, Diffusion Map,
Laplacian Eigenmaps, UMAP, PHATE, MDS, and Kernelized-PCA. Visualizations for
each of these can be found in Appendix A. Additionally, we demonstrate that
diffusion-based embeddings generated using RF-GAP proximities better retain
the random forest’s learning in low dimensions, as shown in Figure A.1.

##  IV Random Forest-Based Manifold Learning

Local connectivity is encoded via a distance metric. A kernel function, (e.g.,
Gaussian kernel) can be applied to the graph distances to provide a local
measure of similarity between observations from which global relationships can
be learned. For example, in diffusion processes, a stochastic matrix, or
diffusion operator, is formed by row-normalizing the pair-wise similarities.
The global structure is learned by powering the diffusion operator, simulating
a random walk between observations.

The quality of learned embedding is highly dependent on the kernel
construction as well as global-structure mapping. Unlike an unsupervised
kernel function, random forest proximities form noise-resilient, locally-
adaptive neighborhoods ensuring that subsequent embeddings are constructed in
a manner relevant to and consistent with the data labels. Similarities between
points of different classes are still reflected in the proximity values,
whereas this inter-class preservation is lost or diminished in class
conditional measures such as the one given by Equation 1. Continuous labels
can also be reflected in the embedding using random forest proximities. We
provide an example in Figure 2. In this figure, embeddings are colored both by
the life expectancy (the target label) as well as the country’s economic
status (developed vs. developing). It is clear that the unsupervised
embeddings create separate clusters for each economic status, while the RF-
PHATE embedding shows a continuum consistent with life expectancy in lower
dimensions.

Figure 2: This figure demonstrates the ability of RF-GAP methods to work with
continuous labels. The figures provided here are two-dimensional embeddings of
the Life Expectancy dataset [20] using UMAP, $t$-SNE, PHATE, and PHATE with
the RF-GAP proximities, colored by life expectancy (top) and country economic
status (bottom). The single most important variable for determining life
expectancy is the country’s economic status: developed or developing. All
three unsupervised models completely separate these labels into two clusters,
while RF-PHATE [21] produces a progression incorporating these discrete labels
in a continuous manner.

##  V Quantifying Results

We use two methods to evaluate the low-dimensional embeddings in the
supervised context. The first approach determines the extent to which the
embedding can be used for the original classification problem. To this end, we
use the 2-dimensional embeddings of unsupervised, supervised, and RF-GAP-based
models to train a $k$-NN classifier to predict the original labels and compare
the accuracy with that of a model trained on the full dataset. All accuracies
were averaged using leave-one-out cross-validation and the overall results
were aggregated across all datasets given in Table B.1. Ideally, the
difference in accuracies should be minimal, that is, we want to retain useful
information without overfitting. In Figure 3, we see that the unsupervised
embeddings produce accuracies lower than those using the full datasets,
demonstrating a loss of information in low dimensions. The class-conditioned,
supervised approaches tend to overfit the labels, generally producing much
higher accuracies (near-perfect, in some examples) as a result of artificial
class separation. The RF-GAP-based embeddings have accuracies more consistent
with models trained on the full dataset, though results vary by method. The
slightly higher values for the majority of RF-GAP methods can be accounted for
by the superior predictive power of random forests over $k$-NN. We show in
Figure A.1 that the $k$-NN predictive accuracy of the RF-GAP embeddings
typically aligns very well with the OOB score of the forest which generated
the proximities, demonstrating the embeddings’ abilities to retain the
forest’s learning in low dimensions.

Figure 3: A $k$-NN model is trained on each embedding and the accuracies are
compared with those of a model trained on the full dataset. Unsupervised
embeddings tend to produce lower accuracies, while supervised embedding
accuracies are inflated. Generally, RF-GAP-based embeddings produce accuracies
more consistent with the full model. All $k$-NN accuracies were assessed using
leave-one-out cross-validation. Score differences are aggregated across 20
random initializations across each dataset provided in Table B.1

The second evaluation technique provides an assessment of the hierarchy of
variable importance retained in the embedding. In a supervised context,
variables that provide higher class-discriminatory power are considered to be
more important. For this evaluation, we first determine a permutation-based
variable importance score using a $k$-NN classification model on the original
supervised task. We then produce a second set of variable importance scores by
regressing onto the embedding using a $k$-NN model trained on the original
dataset. We calculate the correlation between the two variable importance
scores. In Figure 4, we see that supervised models generally outperform
unsupervised models in retaining variable importance, bearing in mind that
class-conditional methods inflate class discrimination. Diffusion-based RF-GAP
methods tend to perform best with this metric, suggesting they better preserve
global, hierarchical variable importance.

Figure 4: The correlation between importance scores shows to what extent
variable importance is retained in the embedding. All RF-GAP methods
outperform each of the unsupervised methods, and most other supervised methods
have higher correlations than supervised methods.

##  VI Conclusion

In this paper, we discussed the weaknesses of existing supervised manifold
learning methods. We showed that RF-GAP-based manifold learning methods
preserve local structure while maintaining global structure relative to data
classes as can be seen in scatterplots of the embeddings. The visual quality
of the RF-GAP embedding depends on the method used, but variable importance is
maintained in low dimensions regardless of the method. Diffusion-based RF-GAP
embeddings tend to retain the random forest’s learning in low dimensions,
suggesting that such methods better maintain the integrity of the kernel from
which the embeddings were derived.

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##  Appendix A Additional Experimental Results

The RF-GAP proximities perfectly preserve the random-forest predictive ability
as was shown in [3]. In Figure A.1, we show that the predictive ability is
well-preserved in low dimensions. In the figure, we see the diffusion-based
methods as well as RFTSNE and RFUMAP methods nearly perfectly preserve the OOB
score when using a $k$-NN classifier on the embeddings.

Figure A.1: The difference between the OOB score from a trained random forest
and the $k$-NN cross-validated score on the low-dimensional embedding. $k$-NN
models trained on the RF-GAP diffusion-based embeddings as well as RFUMAP and
RFTSNE embeddings nearly perfectly match OOB accuracies.

The following figures provide examples to compare unsupervised, supervised,
and RF-GAP-based methods. In most examples, the supervised models create near-
linear separation between classes, while RF-GAP methods preserve relationships
meaningful to the supervised task without global structure disruption.

Figure A.2: 20 compared embeddings on the cars dataset [14]. Embedding types
are depicted by background color: grey for unsupervised, white for class-
conditional supervised, and yellow for RF-GAP methods. Figure A.3: The wine
dataset [22] is simple enough that using labeled information does not add much
to the embedding. At the same time, the RF-GAP methods are not diminished by
the gained information from the random forest. Figure A.4: The sonar dataset
[23] reduced to two dimensions. The unsupervised models fail to find
meaningful patterns, while the non-random forest supervised creates linear
separation between classes. The RF-GAP models meaningfully retain the random
forest model’s learning. Figure A.5: This example of the Parkinson’s dataset
[22] shows that the RF-GAP embeddings provide a somewhat clearer relationship
between the two classes, while the other supervised methods provide perfect
separation and the unsupervised methods contain extra noise.

##  Appendix B Datasets Used

TABLE B.1: This table contains the descriptions datasets used in the quantitative experiments in Figures 3, 4, and A.1. Observations with missing values were removed, and uniquely-identifying variables were also removed. Each dataset was normalized before applying dimensionality reduction. Data | Observations | Variables | Source
---|---|---|---
Banknote | 1372 | 5 | UCI
Breast Cancer | 699 | 16 | UCI
Car | 1728 | 6 | UCI
Diabetes | 678 | 8 | UCI
E. Coli | 336 | 8 | UCI
Glass | 214 | 10 | UCI
Heart Disease | 303 | 13 | UCI
Hill Valley | 606 | 101 | UCI
Ionosphere | 351 | 34 | UCI
Iris | 150 | 4 | UCI
Liver | 345 | 7 | UCI
Lymphography | 148 | 18 | UCI
Optical Digits | 3823 | 64 | UCI
Parkinson’s | 197 | 23 | UCI
Seeds | 210 | 7 | UCI
Sonar | 208 | 60 | UCI
Tic-Tac-Toe | 958 | 9 | UCI
Waveform | 5000 | 21 | UCI
Wine | 178 | 13 | UCI


#  Hyperuniformity of the determinantal point processes
associated with the Heisenberg group 111 This manuscript was prepared for the
Proceedings of the 2021 RIMS (Research Institute for Mathematical Sciences)
Workshop ‘Mathematical Aspects of Quantum Fields and Related Topics’, which
was held online on December 6–8, 2021. The Proceedings will be issued in RIMS
Kôkyûroku edited by Fumio Hiroshima and published at Kyoto University Research
Information Repository and RIMS Homepage.

Makoto Katori 222 Department of Physics, Faculty of Science and Engineering,
Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan; e-mail:
<EMAIL_ADDRESS>
(17 March 2022)

###### Abstract

The Ginibre point process is given by the eigenvalue distribution of a non-
hermitian complex Gaussian matrix in the infinite matrix-size limit. This is a
determinantal point process (DPP) on the complex plane ${\mathbb{C}}$ in the
sense that all correlation functions are given by determinants specified by an
integral kernel called the correlation kernel. Shirai introduced the one-
parameter ($m\in{\mathbb{N}}_{0}$) extensions of the Ginibre DPP and called
them the Ginibre-type point processes. In the present paper we consider a
generalization of the Ginibre and the Ginibre-type point processes on
${\mathbb{C}}$ to the DPPs in the higher-dimensional spaces,
${\mathbb{C}}^{D},D=2,3,\dots$, in which they are parameterized by a
multivariate level $m\in{\mathbb{N}}_{0}^{D}$. We call the obtained point
processes the extended Heisenberg family of DPPs, since the correlation
kernels are generally identified with the correlations of two points in the
space of Heisenberg group expressed by the Schrödinger representations. We
prove that all DPPs in this large family are in Class I of hyperuniformity.

Keywords Hyperuniformity; Ginibre and Ginibre-type point processes;
Determinantal point processes; Extended Heisenberg family of DPPs; Schrödinger
representations of Heisenberg group

##  1 Introduction and Results

We consider the $d$-dimensional Euclid space $\mathbb{R}^{d}$,
$d\in\mathbb{N}:=\\{1,2,\dots\\}$, or the $D$-dimensional complex space
$\mathbb{C}^{D}$, $D\in\mathbb{N}$ as a base space $S$. We assume that $S$ is
associated with a reference measure $\lambda$. We consider an infinite point
process on $S$, which is expressed by an infinite sum of delta measures
concentrated on a set of random points $X_{i},i\in\mathbb{N}$,

$\Xi=\sum_{i:i\in\mathbb{N}}\delta_{X_{i}}.$ (1.1)

We assume that for any bounded domain $\Lambda\subset S$,
$\Xi(\Lambda)<\infty$; that is, accumulation of points does not occur. We also
assume that with respect to the reference measure $\lambda(dx)$ the point
process has a finite density $\rho_{1}(x)<\infty$ at almost every $x\in S$.

We consider a homogeneous point process in the sense that

$\rho_{1}(x)\lambda(dx)={\rm const.}\times dx,\quad x\in S,$

where $dx$ denotes the Lebesgue measure on $S$. The above assumption implies
that for a bounded domain $\Lambda\subset S$,

$\mathbf{E}[\Xi(\Lambda)]\propto\mathrm{vol}(\Lambda).$

Now we consider the number variance in the domain $\Lambda$,

$\mathrm{var}[\Xi(\Lambda)]:=\mathbf{E}[(\Xi(\Lambda)-\mathbf{E}[\Xi(\Lambda)])^{2}],$

which represents local density fluctuation of the point process $\Xi$. The
domain $\Lambda$ is regarded as an observation window to measure the density
fluctuation. If the points are non-correlated and given by a Poisson point
process, then

$\mathrm{var}[\Xi(\Lambda)]\propto\mathrm{vol}(\Lambda).$

Recently in condensed matter physics and related material sciences, correlated
particle systems are said to be in a hyperuniform state when density
fluctuations are anomalously suppressed in large-scale limit. (See, for
instance, [8, 23].) For an infinite random point process $\Xi$, the
hyperuniformity is defined by

$\lim_{\Lambda\to
S}\frac{\mathrm{var}[\Xi(\Lambda)]}{\mathbf{E}[\Xi(\Lambda)]}=0.$

This means that the number variance of points grows more slowly than the
window volume in the limit such that the window covers whole of the space
$\Lambda\to S$.

Torquato [23] proposed three hyperuniformity classes for point processes
concerning asymptotics of number variances. In order to clearly assert this
classification, here we assume that $S=\mathbb{R}^{d}$, $d\in\mathbb{N}$, and
$\Lambda=\mathbb{B}^{(d)}_{R}:=\\{x\in\mathbb{R}^{d}:|x|<R\\}$, $R>0$, where
$\mathrm{vol}(\mathbb{B}^{(d)}_{R})=\pi^{d/2}R^{d}/\Gamma(d/2+1)$ with the
gamma function $\Gamma(z):=\int_{0}^{\infty}e^{-u}u^{z-1}du$,
$\mathrm{Re}\,z>0$. We consider a series of balls with increasing radius $R$,
$\\{\mathbb{B}^{(d)}_{R}\\}_{R>0}$, and the hyperuniform states are classified
as follows;

$\displaystyle\mbox{Class
I}:\qquad\,\,\,\mathrm{var}[\Xi(\mathbb{B}^{(d)}_{R})]\asymp R^{d-1},$
$\displaystyle\mbox{Class
II}:\qquad\,\,\mathrm{var}[\Xi(\mathbb{B}^{(d)}_{R})]\asymp R^{d-1}\log R,$
$\displaystyle\mbox{Class
III}:\qquad\mathrm{var}[\Xi(\mathbb{B}^{(d)}_{R})]\asymp R^{d-\alpha},\quad
0<\alpha<1,\quad\mbox{as $R\to\infty$}.$

Here $f(R)\asymp g(R)$ means that there are finite positive constants $c_{1}$
and $c_{2}$ such that $c_{1}g(R)<f(R)<c_{2}g(R)$. The above characterization
of three classes will be similarly described for any series of windows
$\\{\Lambda_{R}\\}_{R>0}$ labeled by a linear scale $R$ of window. It is
expected that the hyperuniformity and its classification are the proper
properties of $\Xi$ and do not depend on the choice of observation window
$\Lambda_{R}$, $R>0$.

Determinantal point processes (DPPs) studied in random matrix theory (RMT) [7]
provide a variety of examples of hyperuniform systems. In general a DPP is
specified by a triplet

$(\Xi,K,\lambda(dx)),$

where $\Xi$ is a nonnegative-integer-valued Radon measure (1.1) representing a
point process, $K$ is a continuous function $S\times S\to\mathbb{C}$ called
the correlation kernel, and $\lambda(dx)$ is a reference measure on $S$.

The most studied DPP in RMT may be the sinc (sine) DPP, $(\Xi_{\rm
sinc},K_{\rm sinc},dx)$ on $S=\mathbb{R}$ with the correlation kernel

$K_{\rm sinc}(x,y)=\frac{\sin(x-y)}{\pi(x-y)},\quad x,y\in\mathbb{R}.$

This DPP is obtained as the bulk scaling limit of the eigenvalue distribution
of Hermitian random matrices in the Gaussian unitary ensemble (GUE) [7]. If
$\lim_{R\to\infty}f(R)/g(R)=1$, we will write $f(R)\sim g(R)$ as $R\to\infty$.
As a classical result in RMT, it is well known that

$\mathrm{var}[\Xi_{\rm sinc}(\mathbb{B}^{(1)}_{R})]\sim\frac{\log
R}{\pi^{2}}\quad\mbox{as $R\to\infty$}.$

(See, for instance, [5, 20, 21], [19, Remark 5.8].) That is, the sinc DPP is
in Class II of hyperuniformity. Torquato [23] studied one-parameter
($d\in\mathbb{N}$) family of DPPs called the Fermi-sphere point processes,
which is also called the Euclidean family of DPPs in [13]. This family gives
the sinc DPP when $d=1$. It was proved that the Fermi-sphere point processes
are in Class II of hyperuniformity for general $d\in\mathbb{N}$ [24, 23].

An example of infinite DPP in Class I of hyperuniformity is also provided in
RMT [7]. It is the DPP on $\mathbb{C}$ called the Ginibre DPP, $(\Xi_{\rm
Ginibre},K_{\rm Ginibre},\lambda_{\mathrm{N}(0,1;\mathbb{C})}(dx))$ on
$S=\mathbb{C}$, which is obtained as the bulk scaling limit of eigenvalue
distribution of non-Hermitian random matrices [9]. There

$\displaystyle K_{\rm Ginibre}(x,y)$ $\displaystyle=e^{x\overline{y}},\quad
x,y\in\mathbb{C},$ $\displaystyle\lambda_{\mathrm{N}(0,1;\mathbb{C})}(dx)$
$\displaystyle=\frac{e^{-|x|^{2}}}{\pi}dx.$

Note that the disk on $\mathbb{C}$,
$\mathbb{D}_{R}:=\\{x\in\mathbb{C}:|x|<R\\}$, is identified with
$\mathbb{B}^{(2)}_{R}\subset\mathbb{R}^{2}$. Shirai proved [17]

$\mathrm{var}[\Xi_{\rm
Ginibre}(\mathbb{B}^{(2)}_{R})]\sim\frac{R}{\sqrt{\pi}}\quad\mbox{as
$R\to\infty$}.$ (1.2)

In [14] the result (1.2) in $S=\mathbb{C}$ was extended to DPPs in the higher-
dimensional complex spaces $S=\mathbb{C}^{D},D=2,3\dots$ as follows. For
$S=\mathbb{C}^{D},D\in\mathbb{N}$, each coordinate $x\in\mathbb{C}^{D}$ has
$D$ complex components; $x=(x^{(1)},\dots,x^{(D)})$ with
$x^{(\ell)}=\mathrm{Re}\,x^{(\ell)}+\sqrt{-1}\mathrm{Im}\,x^{(\ell)}$,
$\ell=1,\dots,D$. We set
$x_{\mathrm{R}}:=(\mathrm{Re}\,x^{(1)},\dots,\mathrm{Re}\,x^{(D)})$,
$x_{\mathrm{I}}:=(\mathrm{Im}\,x^{(1)},\dots,\mathrm{Im}\,x^{(D)})\in\mathbb{R}^{D}$,
and we write $x=x_{\mathrm{R}}+\sqrt{-1}x_{\mathrm{I}}$ in this paper. The
Lebesgue measure on $\mathbb{C}^{D}$ is given by
$dx=dx_{\mathrm{R}}dx_{\mathrm{I}}:=\prod_{\ell=1}^{D}d\mathrm{Re}\,x^{(\ell)}d\mathrm{Im}\,x^{(\ell)}$.
For $x=x_{\mathrm{R}}+\sqrt{-1}x_{\mathrm{I}}$,
$y=y_{\mathrm{R}}+\sqrt{-1}y_{\mathrm{I}}\in\mathbb{C}^{D}$, we use the
standard Hermitian inner product;

$\displaystyle x\cdot\overline{y}$
$\displaystyle:=(x_{\mathrm{R}}+\sqrt{-1}x_{\mathrm{I}})\cdot(y_{\mathrm{R}}-\sqrt{-1}y_{\mathrm{I}})$
$\displaystyle=(x_{\mathrm{R}}\cdot y_{\mathrm{R}}+x_{\mathrm{I}}\cdot
y_{\mathrm{I}})-\sqrt{-1}(x_{\mathrm{R}}\cdot
y_{\mathrm{I}}-x_{\mathrm{I}}\cdot y_{\mathrm{R}}).$

The norm is given by
$|x|:=\sqrt{x\cdot\overline{x}}=\sqrt{|x_{\mathrm{R}}|^{2}+|x_{\mathrm{I}}|^{2}}$,
$x\in\mathbb{C}^{D}$. Notice that if
$x=x_{\mathrm{R}},y=y_{\mathrm{R}}\in\mathbb{R}^{D}$, then
$x\cdot\overline{y}=x_{\mathrm{R}}\cdot
y_{\mathrm{R}}:=\sum_{\ell=1}^{D}\mathrm{Re}\,x^{(\ell)}\mathrm{Re}\,y^{(\ell)}$.
The $D$-dimensional disk
$\mathbb{D}^{(D)}_{R}:=\\{x\in\mathbb{C}^{D}:|x|<R\\}$ is identified with
$\mathbb{B}^{(d)}_{R}$ in $\mathbb{R}^{d}$ provided that
$d=2D,D\in\mathbb{N}$. On $\mathbb{C}^{D}$ the reference measure is given by
the $D$-dimensional direct-product of
$\lambda_{\mathrm{N}(0,1;\mathbb{C})}(dx)$,

$\displaystyle\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx)$
$\displaystyle:=\prod_{\ell=1}^{D}\lambda_{\mathrm{N}(0,1;\mathbb{C})}(dx^{(\ell)})$
$\displaystyle=\frac{e^{-(|x_{\mathrm{R}}|^{2}+|x_{\mathrm{I}}|^{2})}}{\pi^{D}}dx_{\mathrm{R}}dx_{\mathrm{I}}=\frac{e^{-|x|^{2}}}{\pi^{D}}dx,\quad
x\in\mathbb{C}^{D}.$ (1.3)

In [14] the one-parameter family ($D\in\mathbb{N}$) of DPPs was studied, which
is called the Heisenberg family of DPPs defined on $\mathbb{C}^{D}$ as
follows.

###### Definition 1.1

The Heisenberg family of DPPs is defined by
$(\Xi_{\mathsf{H}_{D}},K_{\mathsf{H}_{D}},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx))$
on $\mathbb{C}^{D}$, $D\in\mathbb{N}$ with the correlation kernel

$K_{\mathsf{H}_{D}}(x,y)=e^{x\cdot\overline{y}},\quad x,y\in\mathbb{C}^{D}.$
(1.4)

Note that $K_{\mathsf{H}_{D}}$ is hermitian;
$\overline{K_{\mathsf{H}_{D}}(x,y)}=K_{\mathsf{H}_{D}}(y,x)$,
$x,y\in\mathbb{C}^{D}$. The kernels in this form on
$\mathbb{C}^{D},D\in\mathbb{N}$ have been studied by Zelditch, who identified
them with the Szegő kernels for the reduced Heisenberg group
$\mathsf{H}_{D}^{\rm red}$ [26]. This family includes the Ginibre DPP as the
lowest dimensional case; $D=1$. The following was proved [14].

###### Theorem 1.2

Any DPP in the Heisenberg family,
$(\Xi_{\mathsf{H}_{D}},K_{\mathsf{H}_{D}},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx))$
on $\mathbb{C}^{D}$, $D\in\mathbb{N}$, is in Class I of hyperuniformity such
that

$\lim_{R\to\infty}R\frac{\mathrm{var}[\Xi_{\mathsf{H}_{D}}(\mathbb{B}^{(2D)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{D}}(\mathbb{B}^{(2D)}_{R})]}=\frac{D}{\sqrt{\pi}}.$

Moreover, for each $D\in\mathbb{N}$, the following asymptotic expansion holds,

$\frac{\mathrm{var}[\Xi_{\mathsf{H}_{D}}(\mathbb{B}^{(2D)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{D}}(\mathbb{B}^{(2D)}_{R})]}\sim\frac{D}{\sqrt{\pi}}R^{-1}\sum_{k=0}^{\infty}(-1)^{k}\frac{\alpha_{k}(D)}{(2k+1)k!2^{4k}}R^{-2k}\quad\mbox{as
$R\to\infty$},$

where

$\alpha_{k}(D)=\begin{cases}1,&\quad\mbox{if
$k=0$},\cr\displaystyle{\prod_{\ell=-k+1}^{k}(2D+2\ell-1)},&\quad\mbox{if
$k\in\mathbb{N}$}.\end{cases}$

Instead of $\mathbb{B}^{(2D)}_{R}\simeq\mathbb{D}^{(D)}_{R}$,
$D\in\mathbb{N}$, we can consider the $D$-dimensional polydisk of radius $R>0$
in $\mathbb{C}^{D}$,

$\Delta^{(D)}_{R}:=\\{x=(x^{(1)},\dots,x^{(D)})\in\mathbb{C}^{D}:|x^{(i)}|<R,i=1,\dots,D\\},$
(1.5)

as an observation window. In Remark 5 of [14], the following was proved.

###### Proposition 1.3

For the Heisenberg family of DPPs,
$(\Xi_{\mathsf{H}_{D}},K_{\mathsf{H}_{D}},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx))$,
$D\in\mathbb{N}$,

$\displaystyle\lim_{R\to\infty}R\frac{\mathrm{var}[\Xi_{\mathsf{H}_{D}}(\Delta^{(D)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{D}}(\Delta^{(D)}_{R})]}$
$\displaystyle=\lim_{R\to\infty}R\left[1-\left(1-\frac{\mathrm{var}[\Xi_{\mathsf{H}_{1}}(\mathbb{B}^{(2)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{1}}(\mathbb{B}^{(2)}_{R})]}\right)^{D}\right]$
$\displaystyle=\frac{D}{\sqrt{\pi}}.$

The Laguerre polynomial is defined by

$L_{n}^{(\alpha)}(\zeta):=\frac{\zeta^{-\alpha}e^{\zeta}}{n!}\frac{d^{n}}{d\zeta^{n}}(\zeta^{n+\alpha}e^{-\zeta}),\quad
n\in\mathbb{N}_{0}:=\\{0,1,\dots\\},\quad\alpha,\zeta\in\mathbb{R}.$ (1.6)

In particuler, we wite $L_{n}(\zeta):=L_{n}^{(0)}(\zeta)$. Note that
$L_{0}^{(\alpha)}(\zeta)=1$ for any $\alpha\in\mathbb{R}$. Let
${}_{3}F_{2}(\alpha_{1},\alpha_{2},\alpha_{3};\beta_{1},\beta_{2};z)$ be the
hypergeometric function defined by

${}_{3}F_{2}(\alpha_{1},\alpha_{2},\alpha_{3};\beta_{1},\beta_{2};z):=\sum_{n=0}^{\infty}\frac{(\alpha_{1})_{n}(\alpha_{2})_{n}(\alpha_{3})_{n}}{(\beta_{1})_{n}(\beta_{2})_{n}}\frac{z^{n}}{n!},$

where $(\alpha)_{n}$ is the Pochhammer symbol;
$(\alpha)_{n}:=\alpha(\alpha+1)\cdots(\alpha+n-1),n\in\mathbb{N}$,
$(\alpha)_{0}:=1$. It is obvious that

${}_{3}F_{2}(\alpha_{1},\alpha_{2},0;\beta_{1},\beta_{2};z)=1.$ (1.7)

For $D=1$, let

$K_{\mathsf{H}_{1}}^{(m)}(x,y)=K_{\mathsf{H}_{1}}(x,y)L_{m}(|x-y|^{2}),\quad
m\in\mathbb{N}_{0},\quad x,y\in\mathbb{C}.$

The one-parameter ($m\in\mathbb{N}_{0}$) family of DPPs,
$(\Xi_{\mathsf{H}_{1}}^{(m)},K_{\mathsf{H}_{1}}^{(m)},\lambda_{\mathrm{N}(0,1;\mathbb{C})})$,
$m\in\mathbb{N}_{0}$ on $\mathbb{C}$ was studied by Shirai [18], who called
the DPPs in this family the Ginibre-type point processes. This family of DPPs
was also studied by Haimi and Hedenmalm [11], where they called the DPPs the
polyanalytic Ginibre ensembles. This family of DPPs on $\mathbb{C}$ has a
physical interpretation in terms of a two-dimensional system of free electrons
in a uniform magnetic field, where the electrons occupy the $m$-th Landau
energy level, $m\in\mathbb{N}_{0}$. Shirai proved the following [18].

###### Proposition 1.4

Any Ginibre-type DPP on $\mathbb{C}$,
$(\Xi_{\mathsf{H}_{1}}^{(m)},K_{\mathsf{H}_{1}}^{(m)},\lambda_{\mathrm{N}(0,1;\mathbb{C})}(dx))$,
$m\in\mathbb{N}_{0}$, is in Class I of hyperuniformity such that

$\lim_{R\to\infty}R\frac{\mathrm{var}[\Xi_{\mathsf{H}_{1}}^{(m)}(\mathbb{B}^{(2)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{1}}^{(m)}(\mathbb{B}^{(2)}_{R})]}=C_{\mathsf{H}_{1}}^{(m)}$

with

$\displaystyle C_{\mathsf{H}_{1}}^{(m)}$
$\displaystyle=\frac{2\Gamma(m+3/2)}{\pi
m!}{{}_{3}F_{2}}\left(-\frac{1}{2},-\frac{1}{2},-m;1,-\frac{1}{2}-m;1\right)$
$\displaystyle\sim\frac{8}{\pi^{2}}m^{1/2}\quad\mbox{as $m\to\infty$}.$ (1.8)

In the present paper, we consider a large family of DPPs which includes both
of the Heisenberg family of DPPs and the Ginibre-type DPPs, which is
parameterized by the dimensionality $D$ of the base space $S=\mathbb{C}^{D}$
and the multivariate level expressed by a set of integers,
$m=(m^{(1)},\dots,m^{(D)})\in\mathbb{N}_{0}^{D}$.

###### Definition 1.5

The extended Heisenberg family of DPPs is defined by
$(\Xi^{(m)}_{\mathsf{H}_{D}}$, $K_{\mathsf{H}^{(m)}_{D}}$,
$\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx))$ on $\mathbb{C}^{D}$,
$D\in\mathbb{N}$ with a multivariate level $m\in\mathbb{N}_{0}^{D}$, where the
correlation kernel is given by

$K^{(m)}_{\mathsf{H}_{D}}(x,y)=K_{\mathsf{H}_{D}}(x,y)\prod_{\ell=1}^{D}L_{m^{(\ell)}}(|x^{(\ell)}-y^{(\ell)}|^{2}),\quad
x,y\in\mathbb{C}^{D}.$ (1.9)

Here $K_{\mathsf{H}_{D}}$ is defined by (1.4).

When $m=0:=(0,\dots,0)$, this family is reduced to the original Heisenberg
family defined by Theorem 1.1. It is obvious that when $D=1$ this family is
identified with the Ginibre-type DPPs of Shirai [18].

In [18] Shirai gave a sufficient condition of Class I of hyperuniformity for
DPPs on $\mathbb{R}^{d}$ with general dimensions $d\geq 1$. There he assumed
that the observation windows are balls $\mathbb{B}^{(d)}_{R}$ and the DPPs are
isotropic in a sense. We can apply this general theorem to the Heisenberg
family of DPPs,
$(\Xi_{\mathsf{H}_{D}},K_{\mathsf{H}_{D}},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx))$
for all $D\in\mathbb{N}$ and the Ginibre-type DPPs on $\mathbb{C}$,
$(\Xi_{\mathsf{H}_{1}}^{(m)},K_{\mathsf{H}_{1}}^{(m)},\lambda_{\mathrm{N}(0,1;\mathbb{C})}(dx))$
for all $m\in\mathbb{N}_{0}$. But, it is not applicable to the extended
Heisenberg family of DPPs on $\mathbb{C}^{D}$,
$(\Xi_{\mathsf{H}_{D}}^{(m)},K_{\mathsf{H}_{D}}^{(m)},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx))$,
if $D\geq 2$ and $m\not=0$, since

$\left|\sqrt{\frac{e^{-|x|^{2}}}{\pi^{D}}}K_{\mathsf{H}_{D}}^{(m)}(x,x^{\prime})\sqrt{\frac{e^{-|x^{\prime}|^{2}}}{\pi^{D}}}\right|^{2}=\frac{1}{\pi^{2D}}e^{-|x-x^{\prime}|^{2}}\prod_{\ell=1}^{D}L_{m^{(\ell)}}(|x^{(\ell)}-{x^{\prime}}^{(\ell)}|^{2})^{2}$

is not a function of
$|x-x^{\prime}|^{2}=\sum_{\ell=1}^{n}|x^{(\ell)}-{x^{\prime}}^{(\ell)}|^{2}$.
In other words, the DPPs in the extended Heisenberg family with $D\geq 2$ or
$m\not=0$ are not isotropic.

In the present paper, however, we prove the following result for the extended
Heisenberg family of DPPs, when observation windows are given by the
$D$-dimensional polydisks (1.5).

###### Theorem 1.6

For the extended Heisenberg family of DPPs,
$(\Xi_{\mathsf{H}_{D}}^{(m)},K_{\mathsf{H}_{D}}^{(m)},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx))$,
$D\in\mathbb{N}$, $m\in\mathbb{N}_{0}^{D}$,

$\displaystyle\lim_{R\to\infty}R\frac{\mathrm{var}[\Xi_{\mathsf{H}_{D}}^{(m)}(\Delta^{(D)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{D}}^{(m)}(\Delta^{(D)}_{R})]}$
$\displaystyle=\lim_{R\to\infty}R\left[1-\prod_{\ell=1}^{D}\left(1-\frac{\mathrm{var}[\Xi_{\mathsf{H}_{1}}^{(m^{(\ell)})}(\mathbb{B}^{(2)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{1}}^{(m^{(\ell)})}(\mathbb{B}^{(2)}_{R})]}\right)\right]$
$\displaystyle=\sum_{\ell=1}^{D}C_{\mathsf{H}_{1}}^{(m^{(\ell)})}.$ (1.10)

By (1.7), (1.8) gives

$C_{\mathsf{H}_{1}}^{(0)}=\frac{2\Gamma(3/2)}{\pi}=\frac{1}{\sqrt{\pi}}.$

Hence, this result can be regarded as a generalization of Proposition 1.3.

In Section 2, we show the relationship between the extended Heisenberg family
of DPPs and the Schrödinger representations of the Heisenberg group
$\mathsf{H}_{D}$, $D\in\mathbb{N}$. There the function denoted by $g$
specifies the representation. The correlation kernels
$K_{\mathsf{H}_{D}}^{(m)}$, $D\in\mathbb{N}$, $m\in\mathbb{N}_{0}^{D}$ are
identified with the correlations of two points in the space of
$\mathsf{H}_{D}$ expressed using the Schrödinger operators acting on $g$. In
Section 3, after giving preliminaries for point processes and comments on
Theorem 1.2, the proof of Theorem 1.6 is given.

By Abreu and his coworkers [2, 3], the DPPs associated with the Schrödinger
representations of $\mathsf{H}_{D}$ are named as the Weyl–Heisenberg
ensembles, in which $g$ is called a window function in the context of the
time-frequency analysis [10]. See also Section 2.6 of [13]. It was proved by
Abreu et al. [4] that the Weyl–Heisenberg ensembles are in the hyperuniform
state of Class I for a general class of window functions.

##  2 Representations of the Heisenberg Group and Correlation Kernels

###  2.1 Schrödinger representations

We briefly review the representation theory of the Heisenberg group [6, 22,
10] in order to explain the reason why we call the DPPs defined by Definition
1.1 the Heisenberg family of DPPs and why we think that the DPPs defined by
Definition 1.5 form its extended family.

Consider the classical and quantum kinetics of a single particle moving in
$\mathbb{R}^{D},D\in\mathbb{N}$. We note that, if $D=3k,k\in\mathbb{N}$, this
represents a $k$-particle system in the three dimensional Euclidean space. The
phase space is given by $\mathbb{R}^{2D}$ with coordinates

$(p,q)=(p_{1},\dots,p_{D},q_{1},\dots,q_{D}).$

In order to describe the Heisenberg Lie algebra $\mathsf{h}_{D}$, we consider
$\mathbb{R}^{2D+1}$ with coordinates
$(p,q,\tau)=(p_{1},\dots,p_{D},q_{1},\dots,q_{D},\tau)$, in which a Lie
bracket is given by

$[(p,q,\tau),(p^{\prime},q^{\prime},\tau^{\prime})]=(0,0,p\cdot
q^{\prime}-q\cdot p^{\prime})=(0,0,[(p,q),(p^{\prime},q^{\prime})]).$

The symplectic form of the Lie bracket $[(p,q),(p^{\prime},q^{\prime})]=p\cdot
q^{\prime}-q\cdot p^{\prime}$ comes from the Poisson bracket in the classical
mechanics and the commutator $[A,B]:=AB-BA$ in quantum mechanics. The
Heisenberg group $\mathsf{H}_{D}$ is the Lie group on $\mathbb{R}^{2D+1}$
satisfying the group law

$ZZ^{\prime}=Z+Z^{\prime}+\frac{1}{2}[Z,Z^{\prime}],\quad
Z,Z^{\prime}\in\mathbb{R}^{2D+1};$

that is,

$(p,q,\tau)(p^{\prime},q^{\prime},\tau^{\prime})=\left(p+p^{\prime},q+q^{\prime},\tau+\tau^{\prime}+\frac{1}{2}(p\cdot
q^{\prime}-q\cdot p^{\prime})\right).$

Let $L^{2}(\mathbb{R}^{D})$ be the set of square integrable functions on
$\mathbb{R}^{D}$, where the inner product is given by

$\langle
f,g\rangle_{L^{2}(\mathbb{R}^{D})}:=\int_{\mathbb{R}^{D}}f(\zeta)\overline{g(\zeta)}d\zeta,\quad
f,g\in L^{2}(\mathbb{R}^{D})$

with the norm $\|f\|_{L^{2}(\mathbb{R}^{D})}:=\sqrt{\langle
f,f\rangle_{L^{2}(\mathbb{R}^{D})}}$, $f\in L^{2}(\mathbb{R}^{D})$, where
$\zeta=(\zeta^{(1)},\dots,\zeta^{(D)})\in\mathbb{R}^{D}$ and $d\zeta$ denotes
the Lebesgue measure on $\mathbb{R}^{D}$. For a smooth function $f$, we
introduce operators $X^{(\ell)}$ and $\mathcal{D}^{(\ell)}$ defined by

$(X^{(\ell)}f)(\zeta)=\zeta^{(\ell)}f(\zeta),\quad(\mathcal{D}^{(\ell)}f)(\zeta)=\frac{1}{2\sqrt{-1}}\frac{\partial
f}{\partial\zeta^{(\ell)}}(\zeta),\quad\ell=1,\dots,D.$

They satisfy the commutation relations

$[X^{(\ell)},\mathcal{D}^{(\ell^{\prime})}]=\frac{\sqrt{-1}}{2}\delta_{\ell\ell^{\prime}},\quad\ell,\ell^{\prime}=1,\dots,D.$

Note that the above will represent the canonical commutation relations in
quantum mechanics,
$[Q^{(\ell)},P^{(\ell^{\prime})}]=\sqrt{-1}\hbar\delta_{\ell\ell^{\prime}}$.
Here we should claim that the value of the Planck constant $\hbar$ is
specially chosen to be 1/2. This corresponds to the choice of the reference
measure on $\mathbb{C}^{D}$ as (1.3). We consider a map from $\mathsf{H}_{D}$
to the group of unitary operators acting on $L^{2}(\mathbb{R}^{D})$ defined by

$\rho(p,q,\tau)=e^{2\sqrt{-1}(p\cdot\mathcal{D}+q\cdot X+\tau I)},$

where $\mathcal{D}:=(\mathcal{D}^{(1)},\dots,\mathcal{D}^{(D)})$,
$X:=(X^{(1)},\dots,X^{(D)})$ and $I$ denotes the identity operator. By the
Baker–Campbell–Hausdorff formula, we can show that

$\rho(p,q,\tau)f(\zeta)=e^{2\sqrt{-1}(\tau+q\cdot\zeta+p\cdot
q/2)}f(\zeta+p),\quad f\in L^{2}(\mathbb{R}^{D}).$

The map $\rho$ is called the Schrödinger representation of $\mathsf{H}_{D}$.
The kernel of $\rho$ is $\\{(0,0,k\pi):k\in\mathbb{Z}\\}$, since $e^{2\pi
k\sqrt{-1}}=1,k\in\mathbb{Z}$. The reduced Heisenberg group
$\mathsf{H}_{D}^{\rm red}$ is defined by $\mathsf{H}_{D}^{\rm
red}:=\mathsf{H}_{D}/\\{(0,0,k\pi):k\in\mathbb{Z}\\}$. We consider the matrix-
element functions of $\rho(p,q,\tau)$ at $(f,g)\in L^{2}(\mathbb{R}^{D})^{2}$,

$\displaystyle M_{f,g}(p,q,\tau)$
$\displaystyle:=\langle\rho(p,q,\tau)f,g\rangle_{L^{2}(\mathbb{R}^{D})}=\langle\rho(-p,q,\tau)\overline{g},\overline{f}\rangle_{L^{2}(\mathbb{R}^{D})}$
$\displaystyle=e^{2\sqrt{-1}\tau}\int_{\mathbb{R}^{D}}e^{2\sqrt{-1}q\cdot\zeta}f\left(\zeta+\frac{p}{2}\right)\overline{g\left(\zeta-\frac{p}{2}\right)}d\zeta,\quad
f,g\in L^{2}(\mathbb{R}^{2}),$ (2.1)

which is also called the Fourier–Wigner transform (see, for instance, [6,
Section 1.4]).

###  2.2 Ground-state representation

We put

$g(\zeta)=\left(\frac{2}{\pi}\right)^{D/4}\frac{e^{-\zeta^{2}}}{\pi^{D/2}}=:G_{0}(\zeta),\quad\zeta\in\mathbb{R}^{D},$
(2.2)

and define the complex variables

$x=(x^{(1)},\dots,x^{(D)}):=p+\sqrt{-1}q=(p^{(1)}+\sqrt{-1}q^{(1)},\dots,p^{(D)}+\sqrt{-1}q^{(D)})\in\mathbb{C}^{D}.$

Then (2.1) becomes

$M_{f,G_{0}}(p,q,\tau)=e^{2\sqrt{-1}\tau}\mathsf{B}[f](x)\frac{e^{-|x|^{2}/2}}{\pi^{D/2}}$

with

$\mathsf{B}[f](x):=\left(\frac{2}{\pi}\right)^{D/4}\int_{\mathbb{R}^{D}}f(\zeta)e^{2\zeta\cdot
x-\zeta^{2}-x^{2}/2}d\zeta,\quad f\in L^{2}(\mathbb{R}^{D}).$ (2.3)

This integral is called the Bargmann transform. For $f\in
L^{2}(\mathbb{R}^{D})$, this integral converges uniformly for $x$ in any
compact subset of $\mathbb{C}^{D}$, and hence $\mathsf{B}[f]$ is an entire
function on $\mathbb{C}^{D}$. We can prove that for $f_{1},f_{2}\in
L^{2}(\mathbb{R}^{D})$

$\langle
M_{f_{1},G_{0}},M_{f_{2},G_{0}}\rangle_{L^{2}(\mathbb{R}^{2D})}=\langle
f_{1},f_{2}\rangle_{L^{2}(\mathbb{R}^{D})}=\langle\mathsf{B}[f_{1}],\mathsf{B}[f_{2}]\rangle_{L^{2}(\mathbb{C}^{D},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})})},$
(2.4)

where $\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}$ is given by (1.3) and

$\langle
F_{1},F_{2}\rangle_{L^{2}(\mathbb{C}^{D},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})})}:=\int_{\mathbb{C}^{D}}F_{1}(x)\overline{F_{2}(x)}\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx)$

with the norm
$\|F\|_{L^{2}(\mathbb{C}^{D},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})})}:=\sqrt{\langle
F,F\rangle_{L^{2}(\mathbb{C}^{D},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})})}}$.
The Bargmann–Fock space $\mathcal{F}_{D}$ is defined by

$\mathcal{F}_{D}:=\left\\{F:\mbox{$F$ is entire on $\mathbb{C}^{D}$ and
$\|F\|_{L^{2}(\mathbb{C}^{D},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})})}<\infty$}\right\\}.$

The equalities (2.4) imply that the Bargmann transform is an isometry from
$L^{2}(\mathbb{R}^{D})$ into $\mathcal{F}_{D}$. Hence, if
$\\{f_{n}\\}_{n\in\mathbb{N}_{0}}$ is a complete orthonormal system (CONS) of
$L^{2}(\mathbb{R}^{D})$, then
$\\{\mathsf{B}[f_{n}]\\}_{n\in\mathbb{N}_{0}^{D}}$ makes a CONS for
$\mathcal{F}_{D}$.

For $(p,q,\tau),(p^{\prime},q^{\prime},\tau^{\prime})\in\mathbb{R}^{2D+1}$, we
consider the correlation of these two points expressed by the Schrödinger
operators acting on $G_{0}$ such that

${\rm
Cor}_{G_{0}}((p,q,\tau),(p^{\prime},q^{\prime},\tau^{\prime})):=\langle\rho(-p,q,\tau)G_{0},\rho(-p^{\prime},q^{\prime},\tau^{\prime})G_{0}\rangle_{L^{2}(\mathbb{R}^{D})}.$
(2.5)

We take a CONS $\\{f_{n}\\}_{n\in\mathbb{N}_{0}}$ of $L^{2}(\mathbb{R}^{D})$,
where we assume $f_{n}\in\mathbb{R}$, $n\in\mathbb{N}_{0}$. Then the two-point
correlation (2.5) is expanded and expressed using the matrix-element functions
(2.1) as

$\displaystyle{\rm
Cor}_{G_{0}}((p,q,\tau),(p^{\prime},q^{\prime},\tau^{\prime}))$
$\displaystyle=\sum_{n\in\mathbb{N}_{0}}\langle\rho(-p,q,\tau)G_{0},f_{n}\rangle_{L^{2}(\mathbb{R}^{D})}\langle
f_{n},\rho(-p^{\prime},q^{\prime},\tau^{\prime})G_{0}\rangle_{L^{2}(\mathbb{R}^{D})}$
$\displaystyle=\sum_{n\in\mathbb{N}_{0}}\langle\rho(p,q,\tau)f_{n},G_{0}\rangle_{L^{2}(\mathbb{R}^{D})}\langle
G_{0},\rho(p^{\prime},q^{\prime},\tau^{\prime})f_{n}\rangle_{L^{2}(\mathbb{R}^{D})}$
$\displaystyle=\sum_{n\in\mathbb{N}_{0}}M_{f_{n},G_{0}}(p,q,\tau)\overline{M_{f_{n},G_{0}}(p^{\prime},q^{\prime},\tau^{\prime})}$
$\displaystyle=\frac{e^{-(|x|^{2}+|x^{\prime}|^{2})/2}}{\pi^{D}}\sum_{n\in\mathbb{N}_{0}}\mathsf{B}[f_{n}](x)\overline{\mathsf{B}[f_{n}](x^{\prime})}.$
(2.6)

The Hermite polynomials on $\mathbb{R}$ are defined by

$H_{n}(\zeta):=(-1)^{n}e^{\zeta^{2}}\frac{d^{n}}{d\zeta^{n}}e^{-\zeta^{2}},\quad
n\in\mathbb{N}_{0},\quad\zeta\in\mathbb{R},$

and then the Hermite orthonormal functions are given by

$\psi_{n}(\zeta):=\frac{1}{\sqrt{2^{n}n!\sqrt{\pi}}}e^{-\zeta^{2}/2}H_{n}(\zeta),\quad
n\in\mathbb{N}_{0},\quad\zeta\in\mathbb{R};$ (2.7)

$\int_{\mathbb{R}}\psi_{n}(\zeta)\psi_{m}(\zeta)d\zeta=\delta_{nm},\,n,m\in\mathbb{N}_{0}$.
Here we make a slight modification as

$\widetilde{\psi}_{n}(\zeta):=2^{1/4}\psi_{n}(\sqrt{2}\zeta),\quad
n\in\mathbb{N}_{0},\quad\zeta\in\mathbb{R}.$

$\\{\widetilde{\psi}_{n}\\}_{n\in\mathbb{N}_{0}}$ makes a real CONS of
$L^{2}(\mathbb{R})$,
$\int_{\mathbb{R}}\widetilde{\psi}_{n}(\xi)\widetilde{\psi}_{m}(\xi)d\xi=\delta_{nm},\,n,m\in\mathbb{N}_{0}$,
as assumed above.

We extend the above results on $\mathbb{R}$ to $\mathbb{R}^{D}$,
$D\in\mathbb{N}$ by simply considering direct products; for
$n:=(n^{(1)},\dots,n^{(D)})\in\mathbb{N}_{0}^{D}$,
$\zeta:=(\zeta^{(1)},\dots,\zeta^{(D)})\in\mathbb{R}^{D}$,

$\Psi_{n}(\zeta):=\prod_{\ell=1}^{D}\widetilde{\psi}_{n^{(\ell)}}(\zeta^{(\ell)}).$
(2.8)

We can show that

$\displaystyle\mathsf{B}[\Psi_{n}](x)$
$\displaystyle:=\prod_{\ell=1}^{D}\mathsf{B}[\widetilde{\psi}_{n^{(\ell)}}](x^{(\ell)})$
$\displaystyle=\prod_{\ell=1}^{D}\phi_{n^{(\ell)}}(x^{(\ell)})=:\Phi_{n}(x),\quad
n\in\mathbb{N}_{0}^{D},\quad x\in\mathbb{C}^{D},$

where

$\phi_{n}(x):=\frac{x^{n}}{\sqrt{n!}},\quad n\in\mathbb{N}_{0},\quad
x\in\mathbb{C}.$ (2.9)

Hence (2.6) is calculated as

$\displaystyle{\rm
Cor}_{G_{0}}((p,q,\tau),(p^{\prime},q^{\prime},\tau^{\prime}))$
$\displaystyle=\frac{e^{-(|x|^{2}+|x^{\prime}|^{2})/2}}{\pi^{D}}\sum_{n\in\mathbb{N}_{0}^{D}}\Phi_{n}(x)\overline{\Phi_{n}(x^{\prime})}$
$\displaystyle=\frac{e^{-(|x|^{2}+|x^{\prime}|^{2})/2}}{\pi^{D}}\prod_{\ell=1}^{D}\sum_{n^{(\ell)}=0}^{\infty}\frac{1}{n^{(\ell)}!}\Big{(}x^{(\ell)}\overline{{x^{\prime}}^{(\ell)}}\Big{)}^{n^{(\ell)}}$
$\displaystyle=\sqrt{\frac{e^{-|x|^{2}}}{\pi^{D}}}K_{\mathsf{H}_{D}}(x,x^{\prime})\sqrt{\frac{e^{-|x^{\prime}|^{2}}}{\pi^{D}}},\quad
x,x^{\prime}\in\mathbb{C}^{D}.$

By the gauge invariance of DPP (see Lemma 3.2 below), this can be identified
with the correlation kernel $K_{\mathsf{H}_{D}}(x,x^{\prime})$ of the
Heisenberg family of DPPs given by (1.4). Note that this result is obtained
due to the special choice of $g$ given by (2.2), which is also written as

$g(\zeta)=G_{0}(\zeta):=\frac{1}{\pi^{D/2}}\Psi_{0}(\zeta),\quad\zeta\in\mathbb{R}^{D}.$
(2.10)

We think that (2.8) represents the eigenstate in a multivariate level
$n\in\mathbb{N}_{0}^{D}$ of the $D$-dimensional direct-product system of
harmonic oscillators. Since $\Psi_{0}(\zeta)$ gives the ground state, the
above results on the choice (2.10) can be regarded as the ground-state
representation.

###  2.3 Higher-level representations

Now we consider the following general choice of $g$,

$\displaystyle g(\zeta)$
$\displaystyle=G_{m}(\zeta):=\frac{1}{\pi^{D/2}}\Psi_{m}(\zeta),$
$\displaystyle\quad
m=(m^{(1)},\dots,m^{(D)})\in\mathbb{N}_{0}^{D},\quad\zeta=(\zeta^{(1)},\dots,\zeta^{(D)})\in\mathbb{R}^{D},$

which corresponds to the higher-level state of the $D$-dimensional system of
harmonic oscillators. The matrix-element function is then given by

$M_{f,G_{m}}(p,q,\tau):=\langle\rho(p,q,\tau)f,G_{m}\rangle_{L^{2}(\mathbb{R}^{D})}=e^{2\sqrt{-1}\tau}\mathsf{B}_{m}[f](x,\overline{x})\frac{e^{-|x|^{2}/2}}{\pi^{D/2}},$

where

$\displaystyle\mathsf{B}_{m}[f](x,\overline{x})$
$\displaystyle:=\left(\frac{2}{\pi}\right)^{D/4}\frac{1}{\sqrt{2^{m}m!}}\int_{\mathbb{R}^{D}}f(\zeta)e^{2\zeta\cdot
x-\zeta^{2}-x^{2}/2}\mathbf{H}_{m}\left(\sqrt{2}\left(\zeta-\frac{x+\overline{x}}{2}\right)\right)d\zeta$
$\displaystyle=\frac{1}{\sqrt{m!}}e^{|x|^{2}}\frac{\partial^{m}}{\partial
x^{m}}\Big{[}e^{-|x|^{2}}\mathsf{B}[f](x)\Big{]}.$

Here we have used the multivariate notations,

$\displaystyle 2^{m}:=\prod_{\ell=1}^{D}2^{m^{(\ell)}},\quad
m!:=\prod_{\ell=1}^{D}m^{(\ell)}!,\quad\frac{\partial^{m}}{\partial
x^{m}}:=\prod_{\ell=1}^{D}\frac{\partial^{m^{(\ell)}}}{\partial{x^{(\ell)}}^{m^{(\ell)}}},\quad\mathbf{H}_{m}(\xi):=\prod_{\ell=1}^{D}H_{m^{(\ell)}}(\xi^{(\ell)}).$

By the definition (2.3), $\mathsf{B}_{0}[f](x)=\mathsf{B}[f](x)$. The integral
transformation $\mathsf{B}_{m}[f](x,\overline{x})$, $m\in\mathbb{N}_{0}^{D}$
is called the true-$m$-Bargmann transform by Vasilevski [25], the polyanalytic
Bargmann transform by Abreu and Feichtinger [1], and has been studied as the
coherent state transform by Mouayn [15].

We can read from Shirai’s paper [18] that, for (2.9),

$\frac{1}{\sqrt{m!}}e^{|x|^{2}}\frac{\partial^{m}}{\partial
x^{m}}\Big{[}e^{-|x|^{2}}\phi_{n}(x)\Big{]}=\sqrt{\frac{m!}{n!}}L_{m}^{(n-m)}(|x|^{2})x^{n-m},\quad
n,m\in\mathbb{N}_{0},\quad x\in\mathbb{C},$

where $L_{n}^{(\alpha)}$ is defined by (1.6). Hence, we have

$\mathsf{B}_{m}[\Phi_{n}](x,\overline{x})=\prod_{\ell=1}^{D}\sqrt{\frac{m^{(\ell)}!}{n^{(\ell)}!}}L_{m^{(\ell)}}^{(n^{(\ell)}-m^{(\ell)})}(|x^{(\ell)}|^{2})x^{n^{(\ell)}-m^{(\ell)}}$

for
$m=(m^{(1)},\dots,m^{(D)}),n=(n^{(1)},\dots,n^{(D)})\in\mathbb{N}_{0}^{D},\,x=(x^{(1)},\dots,x^{(D)})\in\mathbb{R}^{D}$.
It was also shown in [18] that

$\sum_{n=0}^{\infty}\frac{1}{n!}L_{m}^{(n-m)}(|x|^{2})L_{m}^{(n-m)}(|x^{\prime}|^{2})(x\overline{x^{\prime}})^{n-m}=\frac{1}{m!}e^{x\cdot\overline{x^{\prime}}}L_{m}(|x-x^{\prime}|^{2}),\quad
m\in\mathbb{N}_{0},\quad x,x^{\prime}\in\mathbb{C}.$

For each multivariate level $m\in\mathbb{N}_{0}^{D}$, as an extension of
(2.5), the correlation of two points $(p,q,\tau)$ and
$(p^{\prime},q^{\prime},\tau^{\prime})$ in the space of $\mathsf{H}_{D}$ shall
be given by

${\rm
Cor}_{G_{m}}((p,q,\tau),(p^{\prime},q^{\prime},\tau^{\prime})):=\langle\rho(-p,q,\tau)G_{m},\rho(-p^{\prime},q^{\prime},\tau^{\prime})G_{m}\rangle_{L^{2}(\mathbb{R}^{D})}.$

It is evaluated as

$\displaystyle{\rm
Cor}_{G_{m}}((p,q,\tau),(p^{\prime},q^{\prime},\tau^{\prime}))$
$\displaystyle=\sum_{n\in\mathbb{N}_{0}^{D}}M_{\Phi_{n},G_{m}}(p,q,\tau)\overline{M_{\Phi_{n},G_{m}}(p^{\prime},q^{\prime},\tau^{\prime})}$
$\displaystyle=\frac{e^{-(|x|^{2}+|x^{\prime}|^{2})/2}}{\pi^{D}}\sum_{n\in\mathbb{N}_{0}^{D}}\mathsf{B}_{m}[\Phi_{n}](x,\overline{x})\overline{\mathsf{B}_{m}[\Phi_{n}](x^{\prime},\overline{x^{\prime}})}$
$\displaystyle=\frac{e^{-(|x|^{2}+|x^{\prime}|^{2})/2}}{\pi^{D}}e^{x\cdot\overline{x^{\prime}}}\prod_{\ell=1}^{D}L_{m^{(\ell)}}(|x^{(\ell)}-{x^{\prime}}^{(\ell)}|^{2})$
$\displaystyle=\sqrt{\frac{e^{-|x|^{2}}}{\pi^{D}}}K_{\mathsf{H}_{D}}^{(m)}(x,x^{\prime})\sqrt{\frac{e^{-|x^{\prime}|^{2}}}{\pi^{D}}},\quad
x,x^{\prime}\in\mathbb{C}^{D}.$

By Lemma 3.2, this can be identified with the correlation kernel
$K_{\mathsf{H}_{D}}^{(m)}(x,x^{\prime})$ of the extended Heisenberg family of
DPPs given by (1.9).

##  3 Preliminaries and Proofs

###  3.1 Preliminaries for point processes

The configuration space of point process $\Xi=\Xi(\cdot)$ is given by

$\mathrm{Conf}(S)=\Big{\\{}\xi=\sum_{i}\delta_{x_{i}}:\mbox{$x_{i}\in S$,
$\xi(\Lambda)<\infty$ for all bounded set $\Lambda\subset S$}\Big{\\}}.$

Let $\mathcal{B}_{\rm c}(S)$ be the set of all bounded measurable complex
functions on $S$ of compact support, and for $\xi\in\mathrm{Conf}(S)$ and
$\phi\in\mathcal{B}_{\rm c}(S)$ we set

$\langle\xi,\phi\rangle:=\int_{S}\phi(x)\,\xi(dx)=\sum_{i}\phi(x_{i}).$

Random variables written in this form are generally called linear statistics
of a point process $\Xi$ [7]. For a point process $\Xi$, if there exists a
non-negative measurable function $\rho_{1}$ such that

$\mathbf{E}[\langle\Xi,\phi\rangle]=\int_{S}\phi(x)\rho_{1}(x)\lambda(dx)\quad\forall\phi\in\mathcal{B}_{\rm
c}(S).$

$\rho_{1}$ is called the first correlation function of $\Xi$ with respect to
the reference measure $\lambda(dx)$. By definition, $\rho_{1}(x)$ gives the
density of point at $x\in S$ with respect to $\lambda(dx)$. For
$n\in\mathbb{N}$, from $\xi\in\mathrm{Conf}(S)$ we define

$\xi_{n}:=\sum_{i_{1},\dots,i_{n}:i_{j}\not=i_{k},j\not=k}\delta_{x_{i_{1}}}\cdots\delta_{x_{i_{n}}},$

and denote the $n$-product measure of $\lambda$ as $\lambda^{\otimes
n}(dx_{1}\cdots dx_{n}):=\prod_{i=1}^{n}\lambda(dx_{i})$. For a point process
$\Xi$, if there exists a symmetric, non-negative measurable function
$\rho_{n}$ on $S^{n}$ such that

$\mathbf{E}[\langle\Xi_{n},\phi\rangle]=\int_{S^{n}}\phi(x_{1},\dots,x_{n})\rho_{n}(x_{1},\dots,x_{n})\lambda^{\otimes
n}(dx_{1}\cdots dx_{n})\quad\forall\phi\in\mathcal{B}_{\rm c}(S^{n}),$

then we say that $\rho_{n}$ is the $n$-th correlation function of $\Xi$ with
respect to $\lambda(dx)$.

Determinantal point process (DPP) is defined as follows.

###### Definition 3.1

A point process $\Xi$ on $(S,\mathcal{B}_{\rm c}(S),\lambda(dx))$ is said to
be a DPP with a measurable kernel $K:S\times S\to\mathbb{C}$, if the
correlation functions with respect to $\lambda(dx)$ are given by

$\rho_{n}(x_{1},\dots,x_{n})=\det_{1\leq i,j\leq
n}[K(x_{i},x_{j})]\quad\mbox{for every $n\in\mathbb{N}$ and any
$x_{1},\dots,x_{n}\in S$}.$ (3.1)

The integral kernel $K$ is called the correlation kernel. The DPP is specified
by the triplet $(\Xi,K,\lambda(dx))$.

The following fact is well known.

###### Lemma 3.2

Consider a non-vanishing function $f:S\to\mathbb{C}$. Even if the correlation
kernel $K(x,y)$ is transformed as

$K(x,y)\to K_{f}(x,y):=f(x)K(x,y)\frac{1}{f(y)},\quad x,y\in S,$ (3.2)

all correlation functions (3.1) are the same and hence

$(\Xi,K,\lambda(dx))\stackrel{{\scriptstyle\rm(law)}}{{=}}(\Xi,K_{f},\lambda(dx)).$

The transformation (3.2) is called the gauge transformation and the above
property of DPP is referred to gauge invariance. See, for instance, Lemma 3.8
in Section 3.6 of [12].

If the point process is a DPP, $(\Xi,K,\lambda(dx))$, then

$\displaystyle\mathbf{E}[\langle\Xi,\phi\rangle]$
$\displaystyle=\int_{S}\phi(x)K(x,x)\lambda(dx),$
$\displaystyle\mathrm{var}[\langle\Xi,\phi\rangle]$
$\displaystyle=\frac{1}{2}\int_{S\times
S}|\phi(x)-\phi(y)|^{2}K(x,y)K(y,x)\lambda^{\otimes
2}(dxdy),\quad\phi\in\mathcal{B}_{\rm c}(S).$

In particular, when $\phi$ is the indicator function of a bounded domain
$\Lambda\subset S$; $\phi(x)={\bf 1}_{\Lambda}(x):=1$, if $x\in\Lambda$, and
$:=0$, otherwise,

$\displaystyle\mathbf{E}[\langle\Xi(\Lambda)\rangle]$
$\displaystyle=\int_{\Lambda}K(x,x)\lambda(dx),$
$\displaystyle\mathrm{var}[\langle\Xi(\Lambda)\rangle]$
$\displaystyle=\int_{\Lambda}\int_{S\setminus\Lambda}K(x,y)K(y,x)\lambda(dx)\lambda(dy).$

###  3.2 Comments on Theorem 1.2

The Bessel function of the first kind and the modified Bessel function of the
first kind are defined as

$\displaystyle J_{\nu}(z)$
$\displaystyle:=\left(\frac{z}{2}\right)^{\nu}\sum_{n=0}^{\infty}(-1)^{n}\frac{(z/2)^{2n}}{n!\Gamma(\nu+n+1)},$
$\displaystyle I_{\nu}(z)$
$\displaystyle:=\left(\frac{z}{2}\right)^{\nu}\sum_{n=0}^{\infty}\frac{(z/2)^{2n}}{n!\Gamma(\nu+n+1)},\quad\nu>-1,\quad
z\in\mathbb{C}\setminus(-\infty,0],$

respectively. The following were proved in [14].

###### Proposition 3.3

For the Heisenberg family of DPPs,
$(\Xi_{\mathsf{H}_{D}},K_{\mathsf{H}_{D}},\lambda_{\mathrm{N}(0,1;\mathbb{C}^{D})}(dx))$
on $\mathbb{C}^{D}$, $D\in\mathbb{N}$, the following hold,

$\displaystyle\mathbf{E}[\Xi_{\mathsf{H}_{D}}(\mathbb{B}^{(2D)}_{R})]$
$\displaystyle=\frac{R^{2D}}{D!},$
$\displaystyle\mathrm{var}[\Xi_{\mathsf{H}_{D}}(\mathbb{B}^{(2D)}_{R})]$
$\displaystyle=\frac{2R^{2D}}{(D-1)!}\int_{0}^{\infty}\frac{J_{D}(\kappa
R)^{2}}{\kappa}(1-e^{-\kappa^{2}/4})d\kappa$
$\displaystyle=\frac{R^{2D}e^{-2R^{2}}}{D!}\sum_{n=0}^{D-1}[I_{n}(2R^{2})+I_{n+1}(2R^{2})],\quad
R>0.$

Theorem 1.2 is concluded from the above proposition, if we use the following
asymptotic formula of the modified Bessel functions (see, for instance, [16,
Section 10.17]),

$I_{\nu}(x)\sim\frac{e^{x}}{\sqrt{2\pi
x}}\sum_{k=0}^{\infty}(-1)^{k}\frac{\alpha_{k}(\nu)}{k!2^{3k}}x^{-k},\quad\mbox{as
$x\to\infty$}.$

###  3.3 Proof of Theorem 1.6

For $R>0,n,m\in\mathbb{N}_{0}$, let

$p_{n}^{(R,m)}:=\frac{m!}{n!}\int_{0}^{R^{2}}u^{n-m}e^{-u}|L_{m}^{(n-m)}(u)|^{2}du.$

We introduce a series of random variables
$Y^{(R,m^{(\ell)})}_{n^{(\ell)}}\in\\{0,1\\}$, $m^{(\ell)}\in\mathbb{N}_{0}$,
$n^{(\ell)}\in\mathbb{N}_{0}$, $\ell=1,\dots,D$ such that they are mutually
independent and

$Y^{(R,m^{(\ell)})}_{n^{(\ell)}}\sim\mu^{\rm
Bernoulli}_{p^{(R,m^{(\ell)})}_{n^{(\ell)}}},$

where the right-hand side denotes the Bernoulli measure of probability
$p_{n^{(\ell)}}^{(R,m^{(\ell)})}$. By the general theory of the duality
relations between DPPs [13, Theorem 2.6], we can prove that

$\Xi_{\mathsf{H}_{D}}^{(m)}(\Delta^{(D)}_{R})\stackrel{{\scriptstyle\rm
d}}{{=}}\sum_{n\in\mathbb{N}_{0}^{D}}\prod_{\ell=1}^{D}Y^{(R,m^{(\ell)})}_{n^{(\ell)}}.$

Then it is easy to verify that

$\mathbf{E}[\Xi_{\mathsf{H}_{D}}^{(m)}(\Delta^{(D)}_{R})]=\sum_{n\in\mathbb{N}_{0}^{D}}\prod_{\ell=1}^{D}p_{n^{(\ell)}}^{(R,m^{(\ell)})}=\prod_{\ell=1}^{D}\sum_{k=0}^{\infty}p_{k}^{(R,m^{(\ell)})},$

and

$\displaystyle\mathrm{var}[\Xi_{\mathsf{H}_{D}}^{(m)}(\Delta^{(D)}_{R})]$
$\displaystyle=\mathrm{var}\left[\sum_{n\in\mathbb{N}_{0}^{D}}\prod_{\ell=1}^{D}Y^{(R,m^{(\ell)})}_{n^{(\ell)}}\right]=\sum_{n\in\mathbb{N}_{0}^{D}}\mathrm{var}\left[\prod_{\ell=1}^{D}Y^{(R,m^{(\ell)})}_{n^{(\ell)}}\right]$
$\displaystyle=\sum_{n\in\mathbb{N}_{0}^{D}}\left[\prod_{\ell=1}^{D}p_{n^{(\ell)}}^{(R,m^{(\ell)})}-\left(\prod_{\ell=1}^{D}p_{n^{(\ell)}}^{(R,m^{(\ell)})}\right)^{2}\right]$
$\displaystyle=\prod_{\ell=1}^{D}\sum_{k=0}^{\infty}p_{k}^{(R,m^{(\ell)})}-\prod_{\ell=1}^{D}\sum_{k=0}^{\infty}(p_{k}^{(R,m^{(\ell)})})^{2}.$

Hence, we obtain

$\frac{\mathrm{var}[\Xi_{\mathsf{H}_{D}}^{(m)}(\Delta^{(D)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{D}}^{(m)}(\Delta^{(D)}_{R})]}=1-\prod_{\ell=1}^{D}\frac{\sum_{k=0}^{\infty}(p_{k}^{(R,m^{(\ell)})})^{2}}{\sum_{k=0}^{\infty}p_{k}^{(R,m^{(\ell)})}},\quad
m\in\mathbb{N}_{0}^{D}.$

When $D=1$, the above gives

$\frac{\mathrm{var}[\Xi_{\mathsf{H}_{1}}^{(m)}(\mathbb{B}^{(2)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{1}}^{(m)}(\mathbb{B}^{(2)}_{R})]}\equiv\frac{\mathrm{var}[\Xi_{\mathsf{H}_{1}}^{(m)}(\Delta^{(1)}_{R})]}{\mathbf{E}[\Xi_{\mathsf{H}_{1}}^{(m)}(\Delta^{(1)}_{R})]}=1-\frac{\sum_{k=0}^{\infty}(p_{k}^{(R,m)})^{2}}{\sum_{k=0}^{\infty}p_{k}^{(R,m)}},\quad
m\in\mathbb{N}_{0}.$

where we have used the obvious fact that
$\Delta_{R}^{(1)}=\mathbb{D}_{R}\subset\mathbb{C}$ and it is identified with
$\mathbb{B}^{(2)}_{R}\subset\mathbb{R}^{2}$. Hence the first equality in
(1.10) is proved. The second equality is obtained by Proposition 1.4. The
proof is hence complete.

Acknowledgements The present author would like to thank Tomoyuki Shirai and
Shinji Koshida for useful discussion on this subject. This work was supported
by the Grant-in-Aid for Scientific Research (C) (No.19K03674), (B)
(No.18H01124), (S) (No.16H06338), (A) (No.21H04432) of Japan Society for the
Promotion of Science. It was also supported by the Research Institute for
Mathematical Sciences (RIMS), a Joint Usage/Research Center located in Kyoto
University. The author thanks Fumio Hiroshima, Itaru Sasaki and Hayato Saigo
for organizing the 2021 RIMS Workshop ‘Mathematical Aspects of Quantum Fields
and Related Topics’, which was held online on December 6–8, 2021.

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