# Optimal Transport Reconstruction of Baryon Acoustic Oscillations

Farnik Nikakhtar<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, PA 19104 – USA  Ravi K. Sheth
Center for Particle Cosmology, University of Pennsylvania, Philadelphia, PA
19104 – USA  The Abdus Salam International Center for Theoretical Physics,
Strada Costiera 11, Trieste 34151 – Italy  Bruno Lévy  Universités de
Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France  Roya Mohayaee  Sorbonne
Université, CNRS, Institut d’Astrophysique de Paris, 98bis Bld Arago, 75014
Paris, France

###### Abstract

A model-independent, weighted semi-discrete, fast optimal transport algorithm
to reconstruct the Lagrangian positions of proto-halos from their evolved
Eulerian positions is presented. Tests with state-of-art cosmological
simulations show that the positions of proto-halos are reconstructed
accurately, without having to assume a background cosmology. The algorithm,
which makes use of a mass estimate of the biased tracers, but is robust to
errors in this estimation, recovers the shape and amplitude of the initial
pair correlation function of the tracers, enabling sub-percent precision in
the BAO distance scale that is not tied to a cosmological model. In principle,
our algorithm also allows direct and independent determinations of the bias
factor and the smearing scale, potentially providing new methods for breaking
the degeneracy between the bias factor $b$ and $\sigma_{8}$.

baryon acoustic oscillations, optimal transport theory

###### pacs:

The Baryon Acoustic Oscillations (BAO) are frozen sound waves from the pre-
decoupling era that leave a peak in the 2-point correlation function (2PCF) of
the linear matter distribution Peebles and Yu (1970) on scales of order 140
Mpc. This ‘BAO peak’ provides an important distance scale for measuring the
expansion history of the Universe Eisenstein et al. (2005). However,
gravitational evolution shifts and smears this peak in the late-time baryon
plus dark matter distribution and hence in the distribution of biased tracers
Bharadwaj (1996); Crocce and Scoccimarro (2008); Eisenstein et al. (2007a).
This potentially biases the inferred distance scale, so most BAO analyses aim
to sharpen the peak and remove its shift by undoing the effects of the gravity
Eisenstein et al. (2007b); Padmanabhan et al. (2012). This ‘reconstruction’ of
the density field is feasible because on scales of order 140 Mpc, perturbation
theory based analyses should be accurate. However, it is complicated by the
fact that we only observe biased tracers of the full density field. Therefore,
most density field reconstruction methods make assumptions about the
background cosmology as well as the nature of the bias between the observed
tracers and the dark matter Sarpa et al. (2019); Schmittfull et al. (2017);
Hada and Eisenstein (2018). In what follows, we describe and test a method in
which only the bias must be specified.

Figure 1:  Schema for our fast weighted and model-independent semi-discrete
algorithm: Reconstruction of the Laguerre cells from the present distribution
of biased tracers (halos) and dark matter field particles (dust). The volume
of each Laguerre cell represents the mass of the object (halo or dust) to
which it corresponds. The initial power spectrum/correlation function is
obtained directly from the distribution of the barycenters of the Laguerre
cells; there is no need to make any additional Lagrangian approximation or any
other model-dependent intermediary.

Optimal transport (hereafter OT) is a powerful mathematical framework which
has recently found applications in diverse branches of science Benamou and
Brenier (2000); Léonard (2014); Mondino and Suhr (2018); Peyré and Cuturi
(2018). In astrophysics, OT reconstruction is a deterministic algorithm that
recovers the initial Lagrangian positions ${\bf q}$ of a given final Eulerian
distribution of particles ${\bf x}$ by solving the Monge-Ampère-Kantorovich
(MAK) problem Frisch et al. (2002); Brenier et al. (2003). In semi-discrete OT
reconstruction Levy et al. (2021), the final particle distribution is
considered to have evolved from a smooth, uniform, continuous initial field
(rather than a discrete cartesian grid). In this setting, instead of a single
point, a patch of Lagrangian space, a Laguerre cell, is assigned to each
evolved object. The map that assigns the Laguerre cells to final Eulerian
positions is the unique solution to a semi-discrete optimization problem as
well Brenier (1991). The Laguerre tessellation is specified by a set of
$\psi_{i}$ values that is the unique maximizer of what is known as the
Kantorovich dual Villani (2009); Santambrogio (2015); Lévy and Schwindt
(2018):

$K(\psi)=\sum_{i}\int_{V^{\psi}_{i}}\left[\frac{1}{2}|{\bf x}_{i}-{\bf
q}|^{2}-\psi_{i}\right]\;d^{3}q+\sum_{i}v_{i}\psi_{i},$ (1)

subject to $V^{\psi}_{i}\neq\emptyset$ for all $i$. A Laguerre diagram reduces
to the Voronoi diagram van de Weygaert (1994) for identical $\psi_{i}$.
Strictly speaking, the above expression assumes that the Lagrangian density
field is uniform, so $v_{i}$ is the volume of the $i$th Laguerre cell, imposed
as a constraint. The $\psi_{i}$ coefficients correspond to the Lagrange
multiplier associated with the constraint. Clearly, for this to be valid, the
sum over all $v_{i}$ neeeds to be equal to the total volume. Previous semi-
discrete OT reconstructions were limited to using the same $v_{i}$ for all
cells Levy et al. (2021). In contrast, the weighted semi-discrete OT we use
here allows a different $v_{i}$ for each Laguerre cell (see Figure 1).

The objective function $K(\psi)$ is concave – this guarantees the existence
and uniqueness of $\psi$ – and smooth ($C^{2}$). Therefore, the exhaustive
combinatorial computation of the discrete-discrete Monge problem can be
replaced with an efficient and convergent Newton method Kitagawa et al.
(2016); Aurenhammer et al. (1992); Lévy (2015). Hence, our semi-discrete
approach scales as $\mathcal{O}\left(N_{\rm LC}\log N_{\rm LC}\right)$ Levy et
al. (2021), where $N_{\rm LC}$ is the number of Laguerre cells: a significant
improvement over previous algorithms.

The dark matter field is comprised of equal mass particles. The assumption of
uniform initial conditions implies that all the $v_{i}$ in equation 1 are the
same. Ref. Levy et al. (2021) show that the associated OT algorithm accurately
recovers the set of displacements ${\bf x}-{\bf q}$ which maps the $z=0$
position ${\bf x}$ of each (equal-mass) dark matter particle to the barycenter
of its associated Laguerre cell ${\bf q}$.

There are two reasons why treating biased tracers, and halos in particular, is
more complicated.

  * i.

The first is that halos typically span a range of masses.

Accommodating this turns out to be straightforward. The assumption of uniform
initial conditions means that one simply sets $v_{i}\propto m_{i}$ in equation
(1), where $m_{i}=n_{i}\,m_{p}$ is the mass of each object and $m_{p}$ is the
particle mass.

  * ii.

The second is more pernicious: Whereas the initial dark matter field was
uniform, the initial proto-halo distribution was not Mo and White (1996);
Sheth and Tormen (1999). Moreover, the Lagrangian-space clustering of the
proto-halo patches depends on their mass. This means that one may not assume
that the (mass-weighted) barycenters of the Laguerre cells of the biased
tracers are uniformly distributed. In principle, this seriously complicates
the OT approach.

One way to address this is by including a model for the ‘dust’ – the mass that
is not associated with the biased tracers – since the initial distribution of
(mass-weighted) tracers plus dust should be uniform. Typically (e.g. for BAO
surveys), the biased tracers account for only $\sim 20\%$ of the total mass
density. Fortunately, our OT algorithm is fast, so the additional
computational load required to reconstruct dust as well as the biased tracers
is offset by the simplicity of the initial condition.

Therefore, in addition to the masses of the Eulerian objects which are
observed, our weighted OT algorithm also requires a guess regarding the
Eulerian spatial distribution of the dust. In what follows, we use a random
subset of the dark matter particles which are not assigned to the biased
tracers to model the dust. This will be more complicated in real data but we
defer discussion of how to model the dust (using results from, e.g., Cai et
al. (2011); Wang et al. (2014); Paranjape and Alam (2020)), to a follow-up
study.

Figure 2:  Evolved, Eulerian distribution of particles that make up a halo at
$z=0$ (blue dots), and the initial, Lagrangian (i.e. $z=\infty$) distribution
of the same particles (grid-like black dots). Grey shaded region shows the
Laguerre cell associated with each halo, as determined by the weighted-OT
reconstruction. Each panel shows results for a single halo: the halo mass (in
solar units) is shown in the upper left corner. In each panel, yellow and red
crosses show the Eulerian ($z=0$) and Lagrangian ($z=\infty$) center-of-mass
positions. The displacement between the two is what smears and shifts the BAO
feature in the two-point correlation function of these biased tracers. Green
cross shows the barycenter of the Laguerre cell associated with each halo, as
determined by the weighted-OT reconstruction. It is extremely close to the red
cross, indicating that the weighted-OT algorithm describes the displacements
extremely well.

Since the OT reconstructed field is uniform, in all previous (discrete or
semi-discrete) OT work, the reconstructed positions were evolved forward in
time by using ${\bf q}+D(z)\,({\bf x}-{\bf q})$ for some $D(z)\ll 1$, before
spatial statistics were measured. Since $D(z)$ is like a linear theory growth
factor (indeed, this evolution is like the linear Lagrangian ‘Zel’dovich’
approximation), this made the OT-reconstruction appear to be cosmology
dependent, even though OT itself makes no assumptions other than convexity of
the potential flow of matter. In contrast, for biased tracers, the Lagrangian
distribution is non-trivial, so it should be possible to extract the BAO
feature directly from the OT-reconstructed positions ${\bf q}$ of the Laguerre
cells, without needing cosmological perturbation theory.111We show elsewhere
that this can be done even for the dark matter if one estimates the Lagrangian
correlation function by weighting each Laguerre cell by the divergence of
$\mathbf{x-q}$, or by plotting the Laplacian of the correlation function of
the displacement field. In what follows, we demonstrate that our fast,
weighted semi-discrete algorithm (a) works well for biased tracers when a good
proxy for the dust is available, and (b) requires no assumption on cosmology –
it is fully model-independent.

We demonstrate our results using halos identified in 20 realizations of the
HADES simulations of Villaescusa-Navarro et al. (2018). Each simulation
follows the gravitational evolution of $512^{3}$ identical particles, each of
mass $m_{p}=6.566\times 10^{11}h^{-1}M_{\odot}$, in a periodic cube of side
$L=1h^{-1}$Gpc in which the background cosmology is flat,
$\Omega_{\Lambda}=1-\Omega_{m}$, with
$(\Omega_{m},\Omega_{b})=(0.3175,0.049)$, and Hubble constant today
$H_{0}=100h$ km s-1Mpc-1, with $h=0.6711$. The initial fluctuation field is
Gaussian so it is completely specified by its power spectrum, $P_{\rm
Lin}(k)$, which is taken from CLASS, with shape $n_{s}=0.9624$ and amplitude
set by $\sigma_{8}=0.833$.

The biased tracers we consider in what follows are ‘halos’ identified in the
$z=0$ output of each box using a friends-of-friends algorithm with linking
length $b=0.2$. We only use halos with more than 20 particles; this
corresponds to a minimum halo mass of $1.3\times 10^{13}h^{-1}M_{\odot}$.
About 75% percent of the particles in the box are not bound up in such halos:
these make up the ‘dust’.

Our OT algorithm takes as input the list of halo masses and positions and some
knowledge about the spatial distribution of the dust. In practice, we randomly
select a subset of the dust particles: If we choose a fraction $p$, then we
would replace the mass of each by $m_{p}/p$. Since halos have at least 20
particles, we only consider $p\geq 0.1$, as this ensures that each dust
particle is less massive than the least massive halo. In the results which
follow, we set $p=1$, although our results are not particularly sensitive to
this choice.

Figure 3:  Left: Unweighted pair correlation functions measured in non-
overlapping bins of width $2h^{-1}$Mpc. Thick orange curve shows $\xi_{\rm
Eul}$ of the $z=0$ halo centers averaged over 20 simulations, and yellow band
shows the rms scatter around this mean. Thick purple curve and error band
shows $\xi_{\rm Lag}$ of the corresponding Lagrangian (i.e. $z=\infty$)
protohalo centers. Thick green curve and error band – which is very similar to
$\xi_{\rm Lag}$ – shows $\xi_{\rm OT}$ measured in our OT reconstructed field.
Smooth dot-dashed curve shows $\xi_{\rm Lin}$, the correlation function of the
dark matter in linear theory. The ‘linear bias factor’ $b=\sqrt{\xi_{\rm
Eul}/\xi_{\rm Lin}}$ is $\approx 1.4$. Right: Same as left, but now objects
are weighted by their mass, so $b=1.9$; dot-dashed curve now shows
$b^{2}\xi_{\rm Lin}$, and the short and long dashed curves show $(1-1/b)^{-2}$
times $\xi_{\rm Lag}$ and $\xi_{\rm OT}$ respectively. The difference in shape
between $\xi_{\rm Eul}$ and $b^{2}\xi_{\rm Lin}$ is why reconstruction is
necessary. The difference in shape between $\xi_{\rm Lag}$ and $b^{2}\xi_{\rm
Lin}$ is due to scale dependent Lagrangian bias.

The list of $n_{i}$ particles which makes up a halo of mass
$m_{i}=n_{i}\,m_{p}$ defines a protohalo patch in the initial conditions. We
refer to the center of mass of this halo at the time it was identified ($z=0$)
as its Eulerian position ${\bf x}_{i}$, and that of the protohalo as the
Lagrangian position ${\bf q}_{i}$ ($z=\infty$; strictly speaking the initial
conditions are at $z_{\rm IC}=99$, but the displacement between $z_{\rm IC}$
and $\infty$ is negligible). Figure 2 shows three representative halos (black
dots) and the protohalo patches (blue dots) from which they evolved. Yellow
and green crosses show the center of mass positions ${\bf x}_{i}$ and ${\bf
q}_{i}$. The offset between the two, ${\bf S}_{i}\equiv{\bf x}_{i}-{\bf
q}_{i}$ is what causes both the shifting and the smearing of the BAO feature
in the halo pair correlation function. The grey shaded region shows the
Laguerre cell associated with each halo, as determined by the weighted-OT
reconstruction; it reconstructs the shape of the protohalo patch quite well.
This is non-trivial, because the long axis of the protohalo becomes the short
axis of the final halo – the two are not simply rescaled versions of one-
another Despali et al. (2013); Ludlow et al. (2014). We discuss the shapes
elsewhere; here, we focus on the positions. The green cross in each panel
shows the barycenter of the Laguerre cell; it is extremely close to the red
cross, indicating that the weighted-OT algorithm describes, hence undoes, the
displacements extremely well. Therefore, we expect it to reconstruct the BAO
feature very well.

Figure 3 compares pair correlation functions $\xi$ averaged over 20 simulation
boxes. In each panel, the solid orange curve shows the average halo
correlation function $\xi_{\rm Eul}(r)$ (i.e. using the Eulerian positions
$|\mathbf{x}_{i}-\mathbf{x}_{j}|$), and the yellow bands around it show the
standard deviation. The purple curve shows a similar analysis of the
Lagrangian protohalos, $\xi_{\rm Lag}$ (which is built from the
$|\mathbf{q}_{i}-\mathbf{q}_{j}|$ pairs), and the green curve shows $\xi_{\rm
OT}$, which is built from the OT-reconstructed Laguerre cell barycenters.

To make the point that our results do not depend on the halo sample, the right
hand panel shows results when we weight each halo by its mass when computing
the pair counts (this weight is the same for the corresponding protohalos and
Laguerre cells, of course). Comparison with the left hand panel shows that
mass-weighting increases the amplitudes of all the correlation functions, as
expected Sheth and Tormen (1999). (We have also checked that the mass-weighted
correlation function of all the particles – i.e. including the ‘dust’ – is
zero. This confirms that, once all the mass is included, the OT algorithm has
indeed converged to a uniform density initial condition.)

The dot-dashed curve in the left hand panel shows $\xi_{\rm Lin}$, the
correlation function of the dark matter field in linear theory. The ‘linear’
‘scale-independent’ bias factor $b\equiv\sqrt{\xi_{\rm Eul}/\xi_{\rm Lin}}$ in
this panel is about 1.4, whereas it is 1.9 in the right hand panel. The most
important point we wish to make is that, whatever the value of $b$, $\xi_{\rm
OT}$ reproduces the shape of $\xi_{\rm Lag}$ very well, despite the fact that
$\xi_{\rm Lag}$ is rather different from $\xi_{\rm Eul}$, in both panels.

We are most interested in whether the various $\xi$ have different shapes,
since differences in amplitude alone will not bias cosmological constraints.
Therefore, the dot-dashed curve in the right hand panel shows $b^{2}\,\xi_{\rm
Lin}$ (with $b=1.9$) rather than $\xi_{\rm Lin}$ itself. This shows clearly
that, compared to the linear theory shape, the BAO feature is smeared out in
$\xi_{\rm Eul}$. The reason why is well understood Bharadwaj (1996); Crocce
and Scoccimarro (2008); Eisenstein et al. (2007a), and is why reconstruction
is necessary.

Similarly, the difference in amplitude between $\xi_{\rm Lag}$ and $\xi_{\rm
Eul}$ is striking but well understood: for tracers of a fixed number density,
such as Eulerian halos, their Lagrangian protohalos and their OT-reconstructed
Laguerre cells, the correlation functions are related:

$\xi_{\rm Eul}\approx b^{2}\,\xi_{\rm Lin}\quad{\rm then}\quad\xi_{\rm
Lag}\approx(b-1)^{2}\,\xi_{\rm Lin}$ (2)

Nusser and Davis (1994); Fry (1996); Mo and White (1996); Sheth and Tormen
(1999). To test this expectation, the short and long dashed curves in the
right hand panel show the result of multiplying $\xi_{\rm Lag}$ and $\xi_{\rm
OT}$ by a scale-independent factor of $(1-1/b)^{-2}$. This indeed brings the
curves into good agreement on scales of $\sim 70h^{-1}$Mpc. However, compared
to the linear theory shape, the BAO feature in $\xi_{\rm Lag}$ is
significantly enhanced. Although the reason for this ‘scale-dependent bias’ is
also well understood Bardeen et al. (1986); Desjacques et al. (2010); Baldauf
and Desjacques (2017), this shape difference is ignored – or simply not
reproduced – by essentially all other reconstruction schemes (e.g. Padmanabhan
et al., 2012; Schmittfull et al., 2017; Hada and Eisenstein, 2018; Sarpa et
al., 2019). In contrast, $\xi_{\rm OT}$ reproduces $\xi_{\rm Lag}$ on these
scales exquisitely.

The results we have shown so far assume that the mass of each tracer is known
perfectly. Our results are robust to factor-of-two or three errors in the mass
estimate. The fidelity and robustness of the OT-reconstructed $\xi_{\rm OT}$
potentially enables a number of novel analyses which we now discuss. We
caution, however, that before they can be applied to real data, we must be
able to treat two additional complications. The first is that, so far, all our
results are in configuration space; we have yet to show that our OT
reconstruction remains accurate when redshift space distorted positions are
used. The second is that we have yet to build a model for the ‘dust’ which
plays an important role in our algorithm. This is why some of the discussion
below is qualitative: a more quantitative analysis would only be justified
after these two complications have been more fully addressed.

The agreement between $\xi_{\rm OT}$ and $\xi_{\rm Lag}$ is a significant
success of the OT reconstruction. However, since $\xi_{\rm OT}$ (like
$\xi_{\rm Lag}$) differs in shape from $\xi_{\rm Lin}$, we now address how
this compromises the original goal, which is to estimate the BAO scale.

There are three approaches to this problem. The traditional approach fits a
fiducial $\xi_{\rm Lin}$ to the reconstructed correlation function (in this
case, $\xi_{\rm OT}$), but as $\xi_{\rm OT}$ and $\xi_{\rm Lin}$ have
different shapes, this might lead to a bias. We believe the reason for the
shape difference is reasonably well understood Bardeen et al. (1986);
Desjacques et al. (2010); Baldauf and Desjacques (2017), so we are in the
process of determining a better fiducial template shape to fit. The second is
motivated by Padmanabhan et al. (2012); Baldauf and Desjacques (2017), and
seeks to correct the shape of $\xi_{\rm OT}$ so that it becomes more like
$\xi_{\rm Lin}$ by using the OT displacements to distort a uniform grid, and
then cross-correlating the result with the OT field. We discuss both these
efforts elsewhere. The third, as we describe below, is a much simpler test
that is not tied to the detailed shape of $\xi_{\rm Lin}$.

The midpoint between the peak and dip scales in $\xi_{\rm Eul}$, $r_{\rm
LP}\equiv(r_{\rm peak}+r_{\rm dip})/2$, is close in scale to $r_{\rm LP}$ in
$\xi_{\rm Lin}$ Anselmi et al. (2016). In addition, Ref.Anselmi et al. (2016)
argued that $r_{\rm LP}$ should be much less sensitive to the scale-dependent
bias which causes the shapes of $\xi_{\rm Lag}$ and $\xi_{\rm Lin}$ to differ.
The right hand panel of Figure 3 shows that although $\xi_{\rm Eul},\xi_{\rm
Lin}$ and $\xi_{\rm OT}$ have different amplitudes around $r_{\rm peak}$ and
$r_{\rm dip}$, they all intersect at a scale that is approximately half-way in
between. I.e., $r_{\rm LP}$ is indeed relatively immune to both scale
dependent bias and evolution, so we will use it to quantify the gain from
using $\xi_{\rm OT}$ rather than $\xi_{\rm Eul}$.

To estimate $r_{\rm LP}$ we fit polynomials, or more carefully chosen basis
functions, to $\xi$ over the range $r=[60,120]h^{-1}$Mpc, and then
differentiate the fit Anselmi et al. (2018); Nikakhtar et al. (2021, 2022).
The peak and LP scales in $\xi_{\rm Eul}$, $(r_{\rm peak},r_{\rm
LP})/h^{-1}$Mpc = (98.5, 91.1), are shifted from their values in $\xi_{\rm
Lin}$, (99.8, 92.7); this 1% shift is why reconstruction was necessary. In
contrast, they are (100.7, 92.6) in both $\xi_{\rm Lag}$ and $\xi_{\rm OT}$.
This $r_{\rm LP}$ is very similar to its value in $\xi_{\rm Lin}$, showing
that $r_{\rm LP}$ is indeed insensitive to scale-dependent bias, and so OT
reconstruction works well: $r_{\rm LP}$ in $\xi_{\rm OT}$ provides a nearly
unbiased constraint on the cosmological distance scale.

Distance scale estimates use the shape of $\xi_{\rm OT}$ but not its
amplitude. In most analyses, this amplitude is proportional to the product of
a bias factor $b$ and the amplitude of fluctuations in the dark matter field,
which is sometimes referred to as $\sigma_{8}$. As we describe below, the fact
that the OT-reconstructed field is so similar to the Lagrangian one provides
three separate ways to break this $b\sigma_{8}$ degeneracy.

The first is more standard: ratios of the two- and three-point correlation
functions yield determinations of bias factors that are independent of
$\sigma_{8}$ Bernardeau et al. (2002). In the Eulerian field these ratios
involve terms coming from perturbation theory, as well as from bias. However,
since our OT-reconstructions are so faithful to the Lagrangian field,
measuring these ratios in the OT field should be both feasible and easy to
interpret. Moreover, combining them with similar measurements in the evolved
field should provide useful constraints on the terms which arise from
evolution.

Our OT reconstructions enable two other novel methods for determining $b$.
First, because $\xi_{\rm Eul}/\xi_{\rm OT}\approx(1-1/b)^{-2}$ on $\sim
50$-$70h^{-1}$Mpc scales, (Figure 3), one could combine measurements over a
range of $r$ by fitting for that multiplicative factor $A$ which brings
$\xi_{\rm Eul}$ and $\xi_{\rm OT}$ into agreement over this range of scales.
Since $A=(1-1/b)^{2}$ this furnishes a direct estimate of $b$ with no
assumptions about the shape or amplitude of $\xi_{\rm Lin}$.

In addition to the shape of $\xi_{\rm OT}$, the OT algorithm also returns a
distribution of displacements. Let $\Sigma_{\rm OT}$ denote the rms of $|{\bf
x}-{\bf q}_{\rm OT}|$.222We note in passing that $\Sigma_{\rm OT}$ can be used
directly in the Laguerre reconstruction algorithm of Ref. Nikakhtar et al.
(2021, 2022) to provide a consistency check of the BAO distance scale. If we
ignore redshift space distortions, this rms is approximately $\int dk\,P_{\rm
Lin}(k)/2\pi^{2}$, which is independent of $b$. Following Ref. Nikakhtar et
al. (2022), let $\Sigma_{\rm obs}^{2}$ denote the result of performing this
integral with the observed power spectrum $P_{\rm Eul}$ in place of $P_{\rm
Lin}$. Since $P_{\rm Eul}\approx b^{2}\,P_{\rm Lin}$, the ratio $\Sigma_{\rm
obs}/\Sigma_{\rm OT}\approx b$.

In summary: Our weighted OT algorithm provides a cosmology independent
reconstruction of the BAO feature in the correlation function of biased
tracers and potentially provides three distinct ways of determining the bias
parameter $b$ independently of $\sigma_{8}$, each with very different
systematics. In forthcoming work, we will extend our algorithm to work with
redshift-space distorted positions (e.g., by incorporating results of Levy et
al. (2021)), and to model the dust (e.g., Cai et al. (2011); Wang et al.
(2014); Paranjape and Alam (2020)), before applying it to real observational
data.

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=: K^{A, {\rm odd}}_{\R \times \S^1(2 \pi \alpha)}(x, x'),
\nonumber\\
\lim_{N \to \infty} K^{\BN}_{\R \times \S^1(2 \pi \alpha)}(x, x')
&= \frac{1}{2} \left\{
\vartheta_2 \left( \frac{i(x+\overline{x^{\prime}})}{2 \pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\vartheta_2 \left( \frac{i(x-\overline{x^{\prime}})}{2 \pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\right\},
\nonumber\\
&=: K^{B}_{\R \times \S^1(2 \pi \alpha)}(x, x')
\nonumber\\
\lim_{N \to \infty} K^{\CN}_{\R \times \S^1(2 \pi \alpha)}(x, x')
&= \frac{1}{2} \left\{
\vartheta_3 \left( \frac{i(x+\overline{x^{\prime}})}{2 \pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\vartheta_3 \left( \frac{i(x-\overline{x^{\prime}})}{2 \pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\right\}
\nonumber\\
&=: K^{C}_{\R \times \S^1(2 \pi \alpha)}(x, x')
\nonumber\\
\lim_{N \to \infty} K^{\DN}_{\R \times \S^1(2 \pi \alpha)}(x, x')
&= \frac{1}{2} \left\{
\vartheta_3 \left( \frac{i(x+\overline{x^{\prime}})}{2 \pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\vartheta_3 \left( \frac{i(x-\overline{x^{\prime}})}{2 \pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\right\}
\nonumber\\
&=: K^{D}_{\R \times \S^1(2 \pi \alpha)}(x, x'),
\label{eqn:kernels_cylinder1}
\end{align}
$x, x' \in \R \times \S^1(2 \pi \alpha)$.

\begin{prop}
\label{thm:infiniteDPP_cylinder}
The five limit kernels (\ref{eqn:kernels_cylinder1})
define the five kinds of infinite DPPs
on the cylinder,
$(\Xi, K^{A, \sharp}_{\R \times \S^1(2 \pi \alpha)}, \lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}))$,
$\sharp=$even, odd, and
$(\Xi$, $K^{R}_{\R \times \S^1(2 \pi \alpha)}$,
$\lambda_{\rN(0, 1/4)}(d x_{\rR})$ $\lambda_{[0, 2 \pi \alpha)}(d x_{\rI}))$,
$R=B, C, D$.
\end{prop}

The particle densities at $x \in D_{2 \pi \alpha} \simeq \R \times \S^1(2 \pi \alpha)$
are obtained from the limit kernels (\ref{eqn:kernels_cylinder1})
by setting $x'=x$.
Since $x+\overline{x}=2 x_{\rR}$, $x-\overline{x}=2 i x_{\rI}$,
the definitions of the Jacobi theta functions (\ref{eqn:theta})
with parity (\ref{eqn:even_odd})
give explicit expressions for them; for instance,
\begin{align*}
\rho_{\R \times \S^1(2 \pi \alpha)}^B(x)
&= \frac{1}{2}
\left\{
\vartheta_2 \left( \frac{i x_{\rR}}{\pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\vartheta_2 \left( \frac{ x_{\rI}}{2 \pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\right\}
\nonumber\\
&= \sum_{n=1}^{\infty}
e^{-(n-1/2)^2/2 \alpha^2}
\Big[ \cosh\{(2n-1) x_{\rR}/\alpha\} - \cos \{(2n-1) x_{\rI}/\alpha\} \Big],
\nonumber\\
\rho_{\R \times \S^1(2 \pi \alpha)}^C(x)
&= \frac{1}{2}
\left\{
\vartheta_3 \left( \frac{i x_{\rR}}{\pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\vartheta_3 \left( \frac{ x_{\rI}}{2 \pi \alpha};
\frac{i}{2 \pi \alpha^2} \right)
\right\}
\nonumber\\
&= \sum_{n=1}^{\infty}
e^{-n^2/2 \alpha^2}
\Big[ \cosh(2n x_{\rR}/\alpha) - \cos (2n x_{\rI}/\alpha) \Big].
\end{align*}
They show that the obtained particle densities are
indeed nonnegative, and that
\[
\rho_{\R \times \S^1(2 \pi \alpha)}^B(0)=0, \quad
\rho_{\R \times \S^1(2 \pi \alpha)}^C(0)
=\rho_{\R \times \S^1(2 \pi \alpha)}^C( i \pi \alpha)=0.
\]

\vskip 0.3cm
\noindent{\bf Remark 6} \,
In [32], the infinite DPPs on a strip in $\C$
were introduced
by taking an anisotropic scaling limit
associated with $N \to \infty$
of the doubly periodic DPPs
(Theorem 3.4 in [32]).
There the limiting correlation kernels
are expressed by the integrals
of products of Jacobi's theta functions.
We have found that,
if we correctly perform Jacobi's imaginary transformations
(\ref{eqn:Jacobi_imaginary}) of the integrands,
the integrals can be calculated
and the results are identified with the correlation kernels
simply given by (\ref{eqn:kernels_cylinder1}).
\vskip 0.3cm

Using the quasi-periodicity of the Jacobi theta functions (\ref{eqn:quasi_periodic2}),
we can show that, for $\sharp={\rm even, odd}$,
\[
\cS_{1/(2 \alpha)} K^{A, \sharp}_{\R \times \S^1(2 \pi \alpha)}(x, x^{\prime})
\sqrt{\frac{2}{\pi}} e^{-(x_{\rR}^2+{x_{\rR}^{\prime}}^2)}
\frac{1}{2 \pi \alpha}
= \frac{e^{i x_{\rI}/\alpha}}{e^{i x_{\rI}'/\alpha}}
K^{A, \sharp}_{\R \times \S^1(2 \pi \alpha)}(x, x^{\prime})
\sqrt{\frac{2}{\pi}} e^{-(x_{\rR}^2+{x_{\rR}^{\prime}}^2)}
\frac{1}{2 \pi \alpha}.
\]
By the gauge invariance, this implies the
shift invariance,
\begin{align}
&\cS_{1/(2 \alpha)}
\left(\Xi, K^{A, \sharp}_{\R \times \S^1(2 \pi \alpha)},
\lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}) \right)
\nonumber\\
& \qquad
\law \left(\Xi, K^{A, \sharp}_{\R \times \S^1(2 \pi \alpha)},
\lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}) \right),
\label{eqn:shift_typeA_cylinder}
\end{align}
$\sharp = \rm{even, odd}$.
Moreover, by the properties (\ref{eqn:half_tau}) of the Jacobi theta functions,
we obtain the equality
\[
\cS_{1/(4 \alpha)} K^{A, {\rm even}}_{\R \times \S^1(2 \pi \alpha)}(x, x^{\prime})
\sqrt{\frac{2}{\pi}} e^{-(x_{\rR}^2+{x_{\rR}^{\prime}}^2)}
\frac{1}{2 \pi \alpha}
= \frac{e^{i x_{\rI}/(2 \alpha)}}{e^{i x_{\rI}'/(2 \alpha)}}
K^{A, {\rm odd}}_{\R \times \S^1(2 \pi \alpha)}(x, x^{\prime})
\sqrt{\frac{2}{\pi}} e^{-(x_{\rR}^2+{x_{\rR}^{\prime}}^2)}
\frac{1}{2 \pi \alpha}.
\]
Hence again by the gauge invariance,
\[
\cS_{1/(4 \alpha)}
\left(\Xi, K^{A, {\rm even}}_{\R \times \S^1(2 \pi \alpha)},
\lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}) \right)
\law
\left(\Xi, K^{A, {\rm odd}}_{\R \times \S^1(2 \pi \alpha)},
\lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}) \right),
\]
that is, the even-limit and the odd-limit of type $A$
are equivalent up to the shift by $1/(4 \alpha)$ in the real-axis direction.

We note that the periodicity $1/(2 \alpha) \in \R$ of
$\left(\Xi, K^{A, \sharp}_{\R \times \S^1(2 \pi \alpha)},
\lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}) \right)$,
$\sharp=$ even, odd, shown by (\ref{eqn:shift_typeA_cylinder})
becomes zero as $\alpha \to \infty$.
Hence, as the $\alpha \to \infty$ limits of these DPPs,
a uniform system of infinite number of points on $\C$ will be obtained.
In order to see such limit transitions, first
we perform Jacobi's imaginary transformations

\begin{lem}
\label{thm:infiniteDPP_cylinder_Jacobi_Imaginary}
The following equalities hold,
\begin{align*}
& K^{A, {\rm even}}_{\R \times \S^1(2 \pi \alpha)}(x, x^{\prime})
\sqrt{\frac{2}{\pi}} e^{-(x_{\rR}^2+{x_{\rR}^{\prime}}^2)}
\frac{1}{2 \pi \alpha}
= \frac{e^{i x_{\rR} x_{\rI}}}{e^{i x^{\prime}_{\rR} x^{\prime}_{\rI}}}
e^{x \overline{x^{\prime}}}
\frac{1}{\pi} e^{-(|x|^2+|x^{\prime}|^2)/2}
\vartheta_0((x+\overline{x^{\prime}}) \alpha; 2 \pi i \alpha^2),
\nonumber\\
& K^{A, {\rm odd}}_{\R \times \S^1(2 \pi \alpha)}(x, x^{\prime})
\sqrt{\frac{2}{\pi}} e^{-(x_{\rR}^2+{x_{\rR}^{\prime}}^2)}
\frac{1}{2 \pi \alpha}
= \frac{e^{i x_{\rR} x_{\rI}}}{e^{i x^{\prime}_{\rR} x^{\prime}_{\rI}}}
e^{x \overline{x^{\prime}}}
\frac{1}{\pi} e^{-(|x|^2+|x^{\prime}|^2)/2}
\vartheta_3((x+\overline{x^{\prime}}) \alpha; 2 \pi i \alpha^2),
\nonumber\\
& K^B_{\R \times \S^1(2 \pi \alpha)}(x, x^{\prime})
\sqrt{\frac{2}{\pi}} e^{-(x_{\rR}^2+{x_{\rR}^{\prime}}^2)}
\frac{1}{2 \pi \alpha}
\nonumber\\
& \quad
= \frac{e^{i x_{\rR} x_{\rI}}}{e^{i x^{\prime}_{\rR} x^{\prime}_{\rI}}}
\Big\{ e^{x \overline{x^{\prime}}}
\vartheta_0( (x+\overline{x^{\prime}}) \alpha; 2 \pi i \alpha^2)
e^{- x \overline{x^{\prime}}}
\vartheta_0( (x-\overline{x^{\prime}}) \alpha; 2 \pi i \alpha^2) \Big\}
\frac{1}{2 \pi} e^{-(|x|^2+|x^{\prime}|^2)/2},
\nonumber\\
& K^C_{\R \times \S^1(2 \pi \alpha)}(x, x^{\prime})
\sqrt{\frac{2}{\pi}} e^{-(x_{\rR}^2+{x_{\rR}^{\prime}}^2)}
\frac{1}{2 \pi \alpha}
\nonumber\\
& \quad
= \frac{e^{i x_{\rR} x_{\rI}}}{e^{i x^{\prime}_{\rR} x^{\prime}_{\rI}}}
\Big\{ e^{x \overline{x^{\prime}}}
\vartheta_3( (x+\overline{x^{\prime}}) \alpha; 2 \pi i \alpha^2)
e^{- x \overline{x^{\prime}}}
\vartheta_3( (x-\overline{x^{\prime}}) \alpha; 2 \pi i \alpha^2) \Big\}
\frac{1}{2 \pi} e^{-(|x|^2+|x^{\prime}|^2)/2},
\nonumber\\
& K^D_{\R \times \S^1(2 \pi \alpha)}(x, x^{\prime})
\sqrt{\frac{2}{\pi}} e^{-(x_{\rR}^2+{x_{\rR}^{\prime}}^2)}
\frac{1}{2 \pi \alpha}
\nonumber\\
& \quad
= \frac{e^{i x_{\rR} x_{\rI}}}{e^{i x^{\prime}_{\rR} x^{\prime}_{\rI}}}
\Big\{ e^{x \overline{x^{\prime}}}
\vartheta_3( (x+\overline{x^{\prime}}) \alpha; 2 \pi i \alpha^2)
e^{- x \overline{x^{\prime}}}
\vartheta_3( (x-\overline{x^{\prime}}) \alpha; 2 \pi i \alpha^2) \Big\}
\frac{1}{2 \pi} e^{-(|x|^2+|x^{\prime}|^2)/2}.
\end{align*}
\end{lem}
By the asymptotics of the Jacobi theta functions (\ref{eqn:theta_asym}),
the following limit transitions are immediately concluded
from the expressions in Lemma \ref{thm:infiniteDPP_cylinder_Jacobi_Imaginary}.
\begin{prop}
\label{thm:alpha_inf}
The following limit transitions
from the four types of infinite DPPs on
$\R \times \S^1(2 \pi \alpha)$
to the three types of Ginibre DPPs on $\C$ are
\begin{align}
\left(\Xi, K^{A, \sharp}_{\R \times \S^1(2 \pi \alpha)},
\lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}) \right)
& \weaka
\Big(\Xi, K^A_{{\rm Ginibre}}, \lambda_{\rN(0, 1; \C)}(d x) \Big),
\quad \sharp={\rm even, odd},
\nonumber\\
\left.
\begin{array}{l}
\displaystyle{
\left(\Xi, K^{B}_{\R \times \S^1(2 \pi \alpha)},
\lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}) \right)
} \cr
\displaystyle{
\left(\Xi, K^{C}_{\R \times \S^1(2 \pi \alpha)},
\lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}) \right)
\end{array}
\right\}
\Big(\Xi, K^C_{{\rm Ginibre}}, \lambda_{\rN(0, 1; \C)}(d x) \Big),
\nonumber\\
\left(\Xi, K^{D}_{\R \times \S^1(2 \pi \alpha)},
\lambda_{\rN(0, 1/4)}(d x_{\rR}) \lambda_{[0, 2 \pi \alpha)}(d x_{\rI}) \right)
\Big(\Xi, K^D_{{\rm Ginibre}}, \lambda_{\rN(0, 1; \C)}(d x) \Big).
\label{eqn:cylinder_Ginibre}
\end{align}
\end{prop}

%%%  SEC5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\SSC
{Examples in Spaces with Arbitrary Dimensions}
\label{sec:example_d_dim}%%%%%%%%
\subsection{Heisenberg family of infinite DPPs on $\C^{d}$}
\label{sec:HeisenbergDPP}

The Ginibre DPP of type A on $\C$ given in Section \ref{sec:Ginibre_ACD}
can be generalized to the DPPs on $\C^d$ for $d \geq 2$.
This generalization was done by [1, 3, 2]
as the Weyl--Heisenberg ensembles of DPP,
but here we derive the DPPs on $\C^d$, $d \in \N$,
following Corollary \ref{thm:main3} given in
Section \ref{sec:orthogonal}.

Let $S_1=\C^d$, $S_2=\Gamma=\R^d$,
\begin{align*}
\lambda_1(d x) & = \prod_{a=1}^d \lambda_{\rN(0, 1; \C)}(d x^{(a)})
= \frac{1}{\pi^d} e^{-|x|^2} = \frac{1}{\pi^d} e^{-(|x_{\rR}|^2+|x_{\rI}|^2)}
\nonumber\\
&=: \lambda_{\rN(0, 1; \C^d)}(d x),
\nonumber\\
\lambda_2(d y) &= \nu(dy)
= \prod_{a=1}^d \lambda_{\rN(0, 1/4)}(d y^{(a)})
=\left( \frac{2}{\pi} \right)^{d/2} e^{-2 |y|^2},
\end{align*}
\[
\psi_1(x, y)
= e^{-(|x_{\rR}|^2-|x_{\rI}|^2)/2 + 2 (x_{\rR} \cdot y + i x_{\rI} \cdot y)},
\quad x=x_{\rR}+i x_{\rI} \in \C^d, \quad y \in \R^d.
\]
We see that
$\Psi_1(x)^2:=\|\psi_1(x, \cdot) \|_{L^2(\R^d, \nu)}^2=e^{|x|^2}$,
$x \in \C^d$. Hence Assumption 3' is satisfied
and then, by Corollary \ref{thm:main3},
we obtain the DPP on $\C^d$ with the correlation kernel,
\begin{align*}
K^{(d)}(x, x')
&= \left( \frac{2}{\pi} \right)^{d/2}
\int_{\R^d} e^{-2 [ |y|^2-\{(x_{\rR}+i x_{\rI}) + (x^{\prime}_{\rR}-i x^{\prime}_{\rI}) \} \cdot y ]}
d y
\nonumber\\
&= \frac{e^{i x_{\rR} \cdot x_{\rI}}}{e^{i x^{\prime}_{\rR} \cdot x^{\prime}_{\rI}}}
K_{\rm Heisenberg}^{(d)}(x, x')
\end{align*}
\[
K^{(d)}_{\rm Heisenberg}(x, x^{\prime})
= e^{x \cdot \overline{x^{\prime}}}, \quad x, x^{\prime} \in \C^d.
\]

The kernels in this form on $\C^d, d \in \N$ have been
studied by Zelditch and his coworkers
(see [57, 14] and references therein),
who identified them with the Szeg\"o kernels
for the reduced Heisenberg group.
Here we call the DPPs associated with
the correlation kernels in this form
the {\it Heisenberg family of DPPs} on $\C^d, d \in \N$.
This family includes the Ginibre DPP of type A as the
lowest dimensional case with $d=1$.

\begin{df}
\label{thm:Heisenberg_class}
The Heisenberg family of DPP on $\C^d, d \in \N$
is defined by
$\Big(\Xi, K^{(d)}_{\rm Heisenberg}, \lambda_{\rN(0, 1; \C^d)}(d x) \Big)$
\[
K^{(d)}_{\rm Heisenberg}(x, x')
=e^{x \cdot \overline{x'}},
\quad x, x \in \C^d.
\]
\end{df}
\vskip 0.3cm
\[
K^{(d)}_{\rm Heisenberg}(x, x) \lambda_{\rN(0, 1; \C^d)}(dx)
= \frac{1}{\pi^d} d x, \quad x \in \C^d,
\]
every DPP in the Heisenberg family
is uniform on $\C^d$ and the density with respect to
the Lebesgue measure $dx$ is given by $1/\pi^d$.

\subsection{Finite DPPs on $\S^d$}
\label{sec:DPP_S_d}

First we recall basic properties of
spherical harmonics on $\S^d$ [41].
For $d \in \N$, let
$\cP=\cP(\R^{d+1})$ be a vector space of all complex-valued
polynomials on $\R^{d+1}$,
and $\cP_k, k \in \N_0$, be its subspaces
consisting of homogeneous polynomials of degree $k$;
$p(x)=\sum_{|\alpha|=k} c_{\alpha} x^{\alpha}$,
$c_{\alpha} \in \C, x=(x^{(1)}, \dots, x^{(d+1)}) \in \R^{d+1}$,
where we have used the notations
$x^{\alpha} := \prod_{a=1}^{d+1} (x^{(a)})^{\alpha_a}$
with $\alpha:=(\alpha_1, \dots, \alpha_{d+1}) \in \N_0^{d+1}$,
$|\alpha| := \sum_{a=1}^{d+1} \alpha_a$.
The vector space of all harmonic functions in $\cP$ is
denoted by $\cH =\{p \in \cP : \Delta p=0\}$
and let $\cH_k = \cH \cap \cP_k$, $k \in \N_0$.

Now we consider a unit sphere in $\R^{d+1}$ denoted by
$\S^{d}$, in which we use  the polar coordinates
for $x=(x^{(1)}, \dots, x^{(d+1)}) \in \S^{d}$,
\begin{align}
x^{(1)} &= \sin \theta_{d} \cdots \sin \theta_2 \sin \theta_1,
\nonumber\\
x^{(a)} &= \sin \theta_{d} \cdots \sin \theta_a \cos \theta_{a-1},
\quad a=2, \dots, d,
\nonumber\\
x^{(d+1)} &= \cos \theta_{d},
\quad \mbox{with $\theta_1 \in [0, 2 \pi), \quad
\theta_a \in [0, \pi], \quad a=2, \dots, d$}.
\label{eqn:coordinate1}
\end{align}
Note that $\|x\|^2_{\R^{d+1}} := \sum_{a=1}^{d+1} {x^{(a)}}^2 =1$.
For $d=2$, if we put
$\theta_1=\pi/2-\varphi$ and $\theta_2=\theta$,
the polar coordinates (\ref{eqn:polar1})
used in Section \ref{sec:sphere_finite} are obtained.
The standard measure on $\S^{d}$ is given
by the {\it Lebesgue area measure} expressed as
\begin{equation}
d \sigma_{d} (x)
= \sin^{d-1} \theta_{d} \sin^{d-2} \theta_{d-1}
\cdots \sin \theta_2 d \theta_1 \cdots d \theta_{d},
\quad x \in \S^{d}.
\label{eqn:sigma}
\end{equation}
The total measure of $\S^{d}$ is calculated as
\begin{equation}
\omega_{d} = \sigma_{d}(\S^{d})
= \frac{2 \pi^{(d+1)/2}}{\Gamma((d+1)/2)}.
\label{eqn:omega}
\end{equation}
We write the space of harmonic polynomials in $\cH_k$
restricted on $\S^{d}$ as
\[
\cY_{(d, k)} = \left\{ h\Big|_{\S^{d}} : h \in \cH_k \right\},
\quad k \in \N_0.
\]
We can see that
\begin{equation}
D(d, k) = \dim \cY_{(d, k)}=
\frac{(d+2k-1)(d+k-2)!}{(d-1)! k!}
=\frac{2}{(d-1)!} k^{d-1}+{\rm o}(k^{d-1}).
\label{eqn:dim}
\end{equation}

Consider an orthonormal basis
$\{Y^{(d,k)}_j \}_{j=1}^{D(d,k)}$ of $\cY_{(d,k)}$
with respect to $d \sigma_{d}$;
\begin{equation}
\langle Y^{(d, k)}_n, Y^{(d,k)}_m \rangle_{L^2(\S^d, d \sigma_d)}
\int_{\S^{d}} Y^{(d, k)}_n(x) \overline{Y^{(d, k)}_m(x)} d \sigma_{d} (x)
=\delta_{nm}, \quad n, m \in \N_0.
\label{eqn:orthonormal}
\end{equation}
Then, if we put
\[
K^{\cY_{(d, k)}}(x, x')
= \sum_{j=1}^{D(d, k)} Y^{(d, k)}_j(x) \overline{Y^{(d, k)}_j(x')},
\quad x' \in \S^{d},
\]
then $\{K^{\cY_{(d, k)}}(x, x')\}_{x, x' \in \S^{d}}$
give the reproducing kernel in $\cY^{(d,k)}$ in the sense that
\[
Y(x') = \int_{\S^{d}} Y(x) \overline{K^{\cY_{(d, k)}}(x, x')} d \sigma_{d}(x),
\quad \forall Y \in \cY_{(d, k)}.
\]

For $\lambda > -1/2$, we define
\[
={_2}F_1 \left( -k, k+2 \lambda; \lambda+ \frac{1}{2} ;
\frac{1-x}{2} \right),
\]
where ${_2}F_1$ denotes the {\it Gauss hypergeometric function},
{_2}F_1(\alpha, \beta, \gamma; z)
:= \sum_{n=0}^{\infty} \{(\alpha)_n (\beta)_n/\gamma)_n \} z^n/n!
$(\alpha)_n := \alpha(\alpha+1) \cdots (\alpha+n-1)
= \Gamma(\alpha+n)/\Gamma(\alpha)$, $n \in \N$,
$(\alpha)_0 := 1$.
Then the following equality is established,
\[
K^{\cY_{(d, k)}}(x, x')
=\frac{D(d, k)}{\omega_{d}} P^{(d-1)/2}_k(x \cdot x'),
\quad x, x' \in \S^{d},
\]
where $\omega_{d}$ and $D(d,k)$ are
given by (\ref{eqn:omega}) and (\ref{eqn:dim}), respectively,
and $x \cdot x' :=\sum_{a=1}^{d+1} x^{(a)} {x'}^{(a)}$.

We see that $K^{\cY_{(d, k)}}(x, x')$ is $\gO(d+1, \R)$-invariant in the sense
\[
K^{\cY_{(d, k)}}(gx, gx')= K^{\cY_{(d, k)}}(x, x'),
\quad \forall g \in \gO(d+1, \R), \quad
\forall x, x' \in \S^{d}.
\]
Let $\{e_1, \dots, e_{d+1}\}$ be the standard basis of
and $L_0$ be
the stabilizer subgroup of
$\SO(d+1, \R)$ at $e_{d+1}$ represented as
\[
L_0 = \left\{
\left( \begin{array}{ll}
A & {\bf 0} \cr
{^{t}{\bf 0}} & 1
\end{array}
\right)
: A \in \SO(d, \R) \right\}.
\]
We define
\[
\cY_{(d, k)}^{L_0} =
\{ Y \in \cY_{(d, k)} : Y(\ell x)=Y(x), \quad
\forall \ell \in L_0, \quad \forall x \in \S^{d} \},
\]
Then $K^{\cY_{(d, k)}} \in \cY_{(d, k)}^{L_0}$.
The function $P^{\lambda}_k(s)$ is called
the {\it ultraspherical polynomial} [42].
The space $\cY_{(d, k)}^{L_0}$ is a one-dimensional vector space
generated by $P_k^{(d-1)/2}(x \cdot e_{d+1})$.
In general, any $L_0$-invariant function is a constant
for each $L_0$-orbit,
$\cO =\{x \in \S^{d} : x \cdot e_{d+1} = a \},
-1 \leq a \leq 1$,
and hence functions in $\cY_{(d, k)}^{L_0}$ is called
the {\it zonal harmonics} of degree $k$.
Note that, when we set
\[
C_k^{\lambda}(x)= \binom{k+2 \lambda-1}{k}
\]
we call $C_k^{\lambda}(s)$
the {\it Gegenbauer polynomial} of degree $k$ [42].

Fix $d \in \N$ and $k \in \N_0$.
Then, if we consider the case that
$S_2=\N$ with
$\lambda_1(d x)=d \sigma_{d}(x)$,
$L^2(\Gamma, \nu)=\ell^2( \{1, \dots, D(d, k)\}) \subset S_2$,
and $\psi_1(x, n)=Y^{(d, k)}_n(x)$,
then (\ref{eqn:orthonormal})
with Remark 3 in Section \ref{sec:OP}
guarantees Assumption 3'.
Hence Corollary \ref{thm:main3} determines
a unique DPP on $\S^{d}$, in which
the correlation kernel is given by [13]
\begin{align*}
K^{\cY_{(d, k)}}(x, x')
&= \frac{D(d,k)}{\omega_{d}} P^{(d-1)/2}_k(x \cdot x')
\nonumber\\
&= \frac{d-1+2k}{(d-1) \omega_{d}} C^{(d-1)/2}_k(x \cdot x').
\end{align*}
It is obvious that the obtained DPP is
rotationally invariant on $\S^d$, since the kernel
$K^{\cY_{(d, k)}}(x, x')$ depend only on the inner product $x \cdot x'$.
The density of points is uniform on $\S^{d}$
and is given with respect to $\sigma_d(dx)$ by
\begin{align*}
\rho^{\cY_{(d, k)}}
&= K^{\cY_{(d, k)}}(x, x)
= \frac{D(d,k)}{\omega_{d}} P^{(d-1)/2}_k(1)
\nonumber\\
&= \frac{D(d, k)}{\omega_{d}}
= \frac{2 k^{d-1}}{(d-1)! \omega_d}+{\rm o}(k^{d-1}),
\end{align*}
where we have used the fact that
={_2}F_1 \left( -k, k+2 \lambda, \lambda+1/2; 0 \right)
= 1, \lambda> - 1/2$ [42].

Next we consider the DPP on $\S^{d}$ for
fixed $d \in \N$ and $L \in \N$ such that
the correlation kernel is given
by the following finite sum [13],
\begin{align}
K^{(N(d, L))}_{{\rm harmonic}(\S^d)}(x, x')
&: = \sum_{k=0}^{L-1} K^{\cY_{(d, k)}}(x, x')
= \frac{1}{\omega_{d}}
\sum_{k=0}^{L-1} D(d, k) P^{(d-1)/2}_k(x \cdot x')
\nonumber\\
&= \frac{1}{\omega_{d}}
\sum_{k=0}^{L-1} \frac{d-1+2k}{d-1} C^{(d-1)/2}_k(x \cdot x'),
\label{eqn:K_sphere2}
\end{align}
where the total number of points on $\S^d$
is given by
\begin{align}
N(d, L) &= \sum_{k=0}^{L-1} D(d, k)
= \frac{2L+d-2}{d} \binom{d+L-2}{L-1}
\nonumber\\
&=\frac{2}{d !} L^d +{\rm o}(L^d).
\label{eqn:N_d_ell}
\end{align}
The DPP $(\Xi, K^{(N)}_{{\rm harmonic}(\S^d)}, d\sigma_d(x))$
is rotationally invariant in $\S^d$ and
is called the {\it harmonic ensemble}
in $\S^d$ with $N$ points by Beltr\'an {\it et al.} [13].
We note the recurrence relation of the Gegenbauer polynomials
(see, Eq.(18.9.7) in [42]),
\[
(n+\lambda) C^{\lambda}_n(x)
=\lambda( C^{\lambda+1}_n(x)-C^{\lambda+1}_{n-2}(x)).
\]
This implies that
\[
\frac{d-1+2k}{d-1}
=C^{(d+1)/2}_k(x)-C^{(d+1)/2}_{k-2}(x), \quad k \geq 2.
\]
Since $C^{\lambda}_0(x)=1, C^{\lambda}_1(x)=2 \lambda x$,
we obtain the following expression for the
correlation kernel,
\[
K^{(N(d, L))}_{{\rm harmonic}(\S^d)}(x, x')
= \frac{1}{\omega_{d}} \left[ C^{(d+1)/2}_{L-1}(x \cdot x')
+ C^{(d+1)/2}_{L-2}(x \cdot x') \right].
\]
If we introduce the {\it Jacobi polynomials}
defined as [42]
\[
P_n^{(\alpha, \beta)}(x)
{_2}F_1 \left(-n, n+\alpha+\beta+1; \alpha+1; \frac{1-x}{2} \right),
\]
and use the contiguous relation,
$(b-a) \, {_2}F_1(a,b ; c; z)
+a \, {_2}F_1(a+1, b; c; z)-b \, {_2}F_1(a, b+1; c; z)=0$,
the above is written as follows [13],
\begin{equation}
K^{(N(d, L))}_{{\rm harmonic}(\S^d)}(x, x')
=\frac{1}{\omega_d} \frac{N(d, L)}{\binom{L+d/2-1}{L-1}}
P^{(d/2, (d-2)/2)}_{L-1}(x \cdot x'),
\label{eqn:K_sphere4}
\end{equation}
where $\binom{L+d/2-1}{L-1}:=\Gamma(L+d/2)/\{(L-1)! \Gamma(d/2+1)\}
=P^{(d/2, (d-2)/2)}_{L-1}(1)$.

In particular, when $d=1$, for
$x=(x^{(1)}, x^{(2)})=(\sin \theta, \cos \theta)$,
$x^{\prime}=({x^{\prime}}^{(1)}, {x^{\prime}}^{(2)})
=(\sin \theta^{\prime}, \cos \theta^{\prime}) \in \S^1 \subset \R^2$,
$\theta, \theta' \in [0, 2 \pi)$, we have
$x \cdot x^{\prime}=\cos(\theta-\theta^{\prime})$
\begin{align}
K^{(N(1, L))}_{{\rm harmonic}(\S^1)}(x, x^{\prime}) d \sigma_1(x)
&= \frac{1}{2 \pi} {_2}F_1 \left(
\frac{1-(2L-1)}{2}, \frac{1+(2L-1)}{2}; \frac{3}{2};
\sin^2 \frac{\theta-\theta^{\prime}}{2} \right) d \theta
\nonumber\\
&= \frac{\sin \{(2L-1)(\theta-\theta^{\prime})/2 \}}
\frac{d \theta}{2 \pi}
\nonumber\\
\frac{d \theta}{2 \pi},
\label{eqn:S1_CUE}
\end{align}
where we have used the fact that
$N(1, L)=2L-1$ given by (\ref{eqn:N_d_ell}).
This verifies the identification of the 1-sphere case
of the present DPP with the CUE,
$(\Xi, K^{\AN}, \lambda_{[0, 2 \pi)}(d \theta))$,
given in Section \ref{sec:Lie_group}.

On the other hand, when $d=2$,
(\ref{eqn:N_d_ell}) gives
$N(2, L)=L^2$ and
\begin{align*}
K^{(N(2, L))}_{{\rm harmonic}(\S^2)}(x, x^{\prime})
&= \frac{L^2}{4 \pi}
{_2}F_1 \left( -L+1, L+1; 2; \frac{1-x \cdot x^{\prime}}{2} \right)
\nonumber\\
&= \frac{N}{4 \pi}
{_2}F_1 \left(
-\sqrt{N}+1, \sqrt{N}+1; 2; \frac{\|x-x^{\prime}\|^2_{\R^3}}{4} \right),
\end{align*}
which is different from $K_{\S^2}^{(N)}(x, x^{\prime})$
given by (\ref{eqn:K_S2}) in Section \ref{sec:sphere_finite}.

\subsection{Euclidean family of infinite DPPs on $\R^{d}$}
\label{sec:EuclideanDPP}

We consider the vicinity of the north pole $e_{d+1}=(0, \dots, 0, 1)$ on $\S^{d}$
and put $\theta_{d} = r/L$,
$r \in [0, \infty)$.
Then the polar coordinates (\ref{eqn:coordinate1}) behave as
\begin{align*}
x^{(1)} &\simeq \frac{r}{L} \sin \theta_{d-1} \cdots \sin \theta_2 \sin \theta_1
=: \frac{1}{L} \widetilde{x}^{(1)},
\nonumber\\
x^{(a)} &\simeq \frac{r}{L} \sin \theta_{d-1} \cdots \sin \theta_k \cos \theta_{a-1}
=: \frac{1}{L} \widetilde{x}^{(a)},
\quad a=2, \dots, d,
\nonumber\\
x^{(d+1)} &\simeq 1-\frac{1}{2} \left(\frac{r}{L}\right)^2.
\end{align*}
In this case, for $x, x' \in \S^{d}$,
\[
x \cdot x'
= \sum_{a=1}^{d+1} x^{(a)} {x'}^{(a)}
= 1-\frac{1}{2 L^2} \|\widetilde{x}-\widetilde{x}'\|_{\R^d}^2 + {\rm o} \left( \frac{1}{L^2} \right),
\quad \mbox{as $L \to \infty$},
\]
where $\widetilde{x}, \widetilde{x}' \in \R^{d}$ and
$\|\cdot\|_{\R^d}$ denotes the Euclidean norm in $\R^{d}$.
Hence we can conclude that
\begin{equation}
x \cdot x' = \cos \left( \frac{r}{L} \right)
+{\rm o} \left( \frac{1}{L^2} \right),
\quad \mbox{with $r := \|\widetilde{x}-\widetilde{x}'\|_{\R^d}$},
\quad \mbox{ as $L \to \infty$}.
\label{eqn:settingA}
\end{equation}
In this limit, the measure on $\S^{d}$
given by (\ref{eqn:sigma}) behaves as
\begin{align*}
d \sigma_{d}(x)
&\simeq \frac{1}{L^{d}}
r^{d-1} \sin^{d-3} \theta_{d-2} \cdots \sin \theta_2 \,
dr d \theta_1 \cdots d \theta_{d-1}
\nonumber\\
&= \frac{1}{L^{d}} d \widetilde{x},
\quad \widetilde{x} \in \R^{d}.
\end{align*}

The following limit is proved for the correlation kernel
$K^{(N(d, L))}_{{\rm harmonic}(\S^d)}$ given by (\ref{eqn:K_sphere2}). %\cite{KS19a}.
\begin{lem}
\label{thm:asymK}
When (\ref{eqn:settingA}) holds,
the limit
\[
k^{(d)}(r) =
\lim_{L \to \infty} \frac{1}{L^{d}} K^{(N(d, L))}_{{\rm harmonic}(\S^d)}(x, x')
\]
exists and has the following expressions,
\begin{align}
\frac{J_{d/2}(r)}{(2 \pi r)^{d/2}},
\label{eqn:lim_K2}
\\
&= \frac{1}{(2 \pi)^{d/2} r^{(d-2)/2}}
\int_0^1 s^{d/2} J_{(d-2)/2}(r s) d s,
\label{eqn:lim_K}
\end{align}
where $J_{\nu}(z)$ is
the Bessel function of the first kind
with index $\nu$ defined by (\ref{eqn:Bessel_function}).
\end{lem}
\noindent{\it Proof} \,
For $d \in \N$ and $r \in (0, \infty)$,
the following formula of Mehler--Heine type
is known (Theorem 8.1.1 in [54]);
for $\alpha, \beta \in \R$,
\[
\lim_{n \to \infty} n^{-\alpha}
P^{(\alpha, \beta)}_{n} \left( \cos \frac{r}{n} \right)
= \left( \frac{r}{2} \right)^{-\alpha}
\]
where the limit is uniform on compact subset of $\C$.
Then under (\ref{eqn:settingA}), (\ref{eqn:K_sphere4}) gives
\[
\lim_{L \to \infty} \frac{1}{L^{d}} K^{(N(d, L))}_{{\rm harmonic}(\S^d)}
(x, x')
= \frac{2}{\omega_d d!} \Gamma(d/2+1)
\left( \frac{2}{r} \right)^{d/2}
\]
By (\ref{eqn:omega}) and the equality
$\Gamma(2z)=\{2^{2z}/(2 \sqrt{\pi})\}
\Gamma(z) \Gamma \left( z+1/2 \right)$,
we can confirm that
$2 \Gamma(d/2+1)/(\omega_d d! )
= 1/\{(2 \sqrt{\pi})^{d} \}$.
Hence (\ref{eqn:lim_K2}) is proved.
By the integral formula (see, for instance, Eq.(10.22.1) in [42]),
\[
\int z^{\nu+1} J_{\nu}(z) dz = z^{\nu+1} J_{\nu+1}(z),
\]
we can derive (\ref{eqn:lim_K}) from (\ref{eqn:lim_K2}).
The proof is complete. \qed
\vskip 0.3cm

This result implies that for each $d \in \N$
we obtain an infinite-dimensional DPP on $\R^{d}$
such that it is uniform and isotropic on $\R^{d}$ and
the correlation kernel is given by
\begin{equation}
K^{(d)}(x, x') = k^{(d)}(\|x-x'\|_{\R^d}),
\quad x, x' \in \R^{d},
\label{eqn:bK1}
\end{equation}
where $k^{(d)}(r)$ is given by (\ref{eqn:lim_K2}) and (\ref{eqn:lim_K}).

We can give the following alternative expression
for $K^{(d)}$.
\begin{lem}
For $d \in \N$, the correlation kernel
$K^{(d)}$ given by (\ref{eqn:bK1}) with (\ref{eqn:lim_K})
is written as
\begin{equation}
K^{(d)}(x, x')
= \frac{1}{(2\pi)^{d}}
\int_{\R^{d}}
{\bf 1}_{\B^d}(y) e^{i (x-x') \cdot y} d y
= \frac{1}{(2\pi)^{d}}
\int_{\B^{d}} e^{i (x-x') \cdot y} d y,
\label{eqn:Fourier}
\end{equation}
where $\B^d$ denotes the unit ball centered at the origin;
$\B^d := \{ y \in \R^d : |y| \leq 1\}$.
\end{lem}
\noindent{\it Proof} \,
The statement is proved for $d=1$ and 2 by direct calculation as follows.
For $d=1$, (\ref{eqn:lim_K}) gives
\[
k^{(1)}(r)=\sqrt{\frac{r}{2 \pi}}
\int_0^1 s^{1/2} J_{-1/2}(rs) ds.
\]
Here we use the equality
$J_{-1/2}(z)=\sqrt{2/(\pi z)} \cos z$.
\[
k^{(1)}(r) =\frac{1}{\pi} \int_0^1 \cos (rs) ds
=\frac{1}{2 \pi} \int_{-1}^{1} e^{i r y} d y,
\]
which gives (\ref{eqn:Fourier}) with $d=1$,
if we put $r=x-x'$ and regard an interval $[-1, 1] \subset \R$
as $\B^1$.
For $d=2$, (\ref{eqn:lim_K}) gives
\begin{equation}
k^{(2)}(r) = \frac{1}{2\pi} \int_0^1 s J_0(rs) ds.
\label{eqn:K3_1}
\end{equation}
We use the following integral representation for
$J_0$ given as Eq.(10.9.1) in [42],
\[
J_0(z)= \frac{1}{\pi} \int_0^{\pi} \cos(z \cos \varphi) d \varphi
=\frac{1}{2 \pi} \int_0^{2 \pi} e^{iz \cos \varphi} d \varphi.
\]
Hence (\ref{eqn:K3_1}) is written as
\begin{equation}
k^{(2)}(r)= \frac{1}{(2 \pi)^2}
\int_0^1 ds \, s \int_0^{2 \pi} d \theta \,
e^{i r s \cos \theta}.
\label{eqn:K3_2}
\end{equation}
We can identify the integral variables $(s, \theta)$ in
(\ref{eqn:K3_2}) with the polar coordinates in $\R^2$
and (\ref{eqn:Fourier}) with $d=2$ is obtained,
if we recognize $r=\|x-x'\|_{\R^2}$,
$s=\|y\|_{\R^2}$, and $(x-x') \cdot y=r s \cos \theta$.
Now we assume $d \geq 3$.
In this case RHS of (\ref{eqn:Fourier}) is given by
\begin{align*}
I &:= \frac{1}{(2\pi)^{d}}
\int_0^1 d s \, s^{d-1}
\int_0^{2 \pi} d \theta_1 \int_0^{\pi} d \theta_2 \, \sin \theta_2
\nonumber\\
& \qquad \times
\cdots \times \int_0^{\pi} d \theta_{d-2} \sin^{d-3} \theta_{d-2}
\int_0^{\pi} d \theta_{d-1} \sin^{d-2} \theta_{d-1}
e^{i r s \cos \theta_{d-1}}.
\end{align*}
\[
\int_0^{2 \pi} d \theta_1 \int_0^{\pi} d \theta_2 \, \sin \theta_2
\cdots \int_0^{\pi} d \theta_{d-2} \sin^{d-3} \theta_{d-2}
=\sigma(\S^{d-2}) = \omega_{d-2},
\]
we have
\[
\int_0^1 d s \, s^{d-1}
\int_0^{\pi} d \theta_{d-1} \, \sin^{d-2} \theta_{d-1}
e^{i r s \cos \theta_{d-1}}.
\]
If we use the following integral representation
of the Bessel function of the first kind,
\[
J_m(z)= \frac{1}{\sqrt{\pi} \Gamma(m+1/2)}
\left( \frac{z}{2} \right)^m
\int_0^{\pi} \sin^{2m}(\theta) e^{iz \cos \theta} d \theta,
\quad m \in \frac{1}{2} \N,
\]
which is obtained from
Eq.(10.9.4) in [42],
the equivalence between (\ref{eqn:Fourier}) and
(\ref{eqn:lim_K}) is verified.
Hence the proof is complete. \qed
\vskip 0.3cm

The kernel (\ref{eqn:Fourier}) is obtained as
the correlation kernel $K_{S_1}$ given by
(\ref{eqn:K_simple2}) in Corollary \ref{thm:main3},
if we consider the case such that
$\lambda_1(d x)=d x$,
$\lambda_2(d y)=\nu(d y)=d y$,
$\psi_1(x, y)=e^{i x \cdot y}/(2\pi)^{d/2}$, and
$\Gamma=\B^d \subsetneq \R^d$.
We see $\Psi_1(x)^2 := \|\psi_1(x, \cdot)\|_{L^2(\Gamma, d \nu)}^2
\equiv |\B^d|/(2 \pi)^d$, $x \in \R^d$,
where the volume of $\B^{d}$ is denoted by
$|\B^d| =\pi^{d/2}/\Gamma((d+2)/2)$.

The kernels $K^{(d)}$ on $\R^d, d \in \N$ have been
studied by Zelditch and others
(see [57, 52, 58, 18] and references therein),
who regarded them as the Szeg\"o kernels
for the reduced Euclidean motion group.
Here we call the DPPs associated with
the correlation kernels in this form
the {\it Euclidean family of DPPs} on $\R^d, d \in \N$.

\begin{df}
\label{thm:EuclideanDPP}
The Euclidean family of DPP on $\R^d, d \in \N$ is defined by
$\Big( \Xi, K^{(d)}_{\rm Euclidean}, d x \Big)$
with the correlation kernel
\begin{align*}
K^{(d)}_{\rm Euclid}(x, x')
&= \frac{1}{(2 \pi)^{d/2}} \frac{J_{d/2}(\|x-x'\|_{\R^d})}{\|x-x'\|_{\R^d}^{d/2}}
\nonumber\\
&=\frac{1}{(2 \pi)^{d/2}}
\frac{1}{\|x-x'\|_{\R^d}^{(d-2)/2}}
\int_0^1 s^{d/2} J_{(d-2)/2} (\|x-x'\|_{\R^d} s) ds
\nonumber\\
&= \frac{1}{(2 \pi)^d}
\int_{\R^d} {\bf 1}_{\B^d}(y)
e^{i (x-x') \cdot y} d y
= \frac{1}{(2 \pi)^d}
\int_{\B^d}
e^{i (x-x') \cdot y} d y,
\quad x, x' \in \R^d.
\end{align*}
\end{df}

The above result is summarized as follows [33].
\begin{prop}
\label{thm:limit_Euclid}
The following is established for $d \in \N$,
\[
\left( \frac{d!}{2} \right)^{1/d}
N^{1/d} \circ \Big( \Xi, K^{(N)}_{{\rm harmonic}(\S^d)}, d \sigma_{d}(x) \Big)
\weak \Big( \Xi, K^{(d)}_{\rm Euclid}, d x \Big).
\]
\end{prop}

We see that
\[
K^{(d)}_{\rm Euclid}(x, x)
=\lim_{r \to 0} \frac{1}{(2\pi)^{d/2}} \frac{J_{d/2}(r)}{r^{d/2}}
=\frac{1}{2^d \pi^{d/2} \Gamma((d+2)/2)}.
\]
Then the Euclidean family of DPP is uniform on $\R^d$
with the density
\[
\rho^{(d)}_{\rm Euclid}= \frac{1}{2^d \pi^{d/2} \Gamma((d+2)/2)}
\]
with respect to the Lebesgue measure $d x$ of $\R^d$.

For lower dimensions, the correlation kernels
and the densities are given as follows,
\begin{align*}
K^{(1)}_{\rm Euclid}(x, x')
&= \frac{\sin(x-x')}{\pi(x-x')}
=K_{\rm sinc}(x, x')
\quad
\mbox{with} \quad \rho^{(1)}_{\rm Euclid} = \frac{1}{\pi},
\nonumber\\
K^{(2)}_{\rm Euclid}(x, x')
&= \frac{J_1(\|x-x'\|_{\R^2})}{2 \pi \|x-x'\|_{\R^2}}
\quad
\mbox{with} \quad \rho^{(2)}_{\rm Euclid} = \frac{1}{4 \pi},
\nonumber\\
K^{(3)}_{\rm Euclid}(x, x')
&= \frac{1}{2 \pi^2 \|x-x'\|_{\R^3}^2}
\left( \frac{\sin \|x-x'\|_{\R^3}}{\|x-x'\|_{\R^3}}
-\cos \|x-x'\|_{\R^3} \right)
\quad
\mbox{with} \quad \rho^{(3)}_{\rm Euclid} = \frac{1}{6 \pi^2}.
\end{align*}
This family of DPPs includes the DPP with the sinc kernel
$K_{\rm sinc}$ as the
lowest dimensional case with $d=1$.
Since $(\Xi, K^{(N)}_{{\rm harmonic}(\S^1)}, d \sigma_1)$
has been identified with the CUE, $(\Xi, K^{\AN}, \lambda_{[0, 2 \pi)})$,
by (\ref{eqn:S1_CUE}), Proposition \ref{thm:limit_Euclid}
can be regarded as the multidimensional extension
of the limit theorem from the CUE to
the DPP with $K_{\rm sinc}$ given by the first line of
Note that,
if $d$ is odd,
\[
=\left(- \frac{1}{2\pi r} \frac{d}{dr} \right)^{(d-1)/2}
\frac{\sin r}{\pi r}.
\]
This is proved by Rayleigh's formula
for the spherical Bessel function of the first kind
(Eq. (10.49.14) in [42]);
\[
j_m(x) := \sqrt{\frac{\pi}{2 x}}
=x^m \left( - \frac{1}{x}
\frac{d}{dx} \right)^m \frac{\sin x}{x},
\quad m \in \N.
\]

%%%  SEC6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\SSC
{Concluding Remarks} \label{sec:remarks}%%%%%%%%

In Section \ref{sec:DPP_S_d}, we studied the finite DPPs
$(\Xi, K^{(N)}_{{\rm harmonic}(\S^d)}, d \sigma_d(x))$, $N \in \N$
called the harmonic ensembles on $\S^d$,
$d \in \N$ [13].
Then we proved as Proposition \ref{thm:limit_Euclid}
in Section \ref{sec:EuclideanDPP}
that their bulk scaling limits are
given by $(\Xi, K_{\rm Euclid}^{(d)}, dx)$,
which we call the Euclidean family of DPPs.
On $\S^2$,
there are two distinct types of uniform and isotropic DPPs,
one of which is the harmonic ensemble
$(\Xi, K^{(N)}_{{\rm harmonic}(\S^2)}, d \sigma_2(x))$
studied in Section \ref{sec:DPP_S_d} [13] ,
and other of which is the DPP called
the spherical ensemble $(\Xi, K^{(N)}_{\S^2}, d \sigma_2(x))$
studied in Section \ref{sec:sphere_finite} [37, 4].
As mentioned above, the scaling limit of
the former is given by $(\Xi, K^{(2)}_{\rm Euclid}, dx)$,
while as given by Proposition \ref{thm:S2_Ginibre}
the bulk scaling limit of the latter is
$(\Xi, K_{\rm Ginibre}^A, \lambda_{\rN(0,1; \C)}(dx))$,
which is equivalent with
$(\Xi, K^{(1)}_{\rm Heisenberg}, \lambda_{\rN(0, 1; \C)}(dx))$.
The spherical ensemble on $\S^2$ shall be generalized
to DPPs on the higher dimensional spheres
$\S^{2d} \simeq \C^d$, $d \geq 2$ so that
they are uniform and isotropic and
their bulk scaling limits are given by
DPPs in the Heisenberg family.
See [11, 12].

With $L^2(S, \lambda)$ and $L^2(\Gamma, \nu)$,
we can consider the system of
{\it biorthonormal functions},
which consists of a pair of distinct families of measurable functions
$\{\psi(x, \gamma) : x \in S, \gamma \in \Gamma \}$
and $\{\varphi(x, \gamma) : x \in S, \gamma \in \Gamma \}$
satisfying the biorthonormality relations
\begin{equation}
\langle \psi(\cdot, \gamma), \varphi(\cdot, \gamma') \rangle_{L^2(S, \lambda)}
\nu(d \gamma)=\delta(\gamma-\gamma') d \gamma,
\quad \gamma, \gamma' \in \Gamma.
\label{eqn:bi_ortho}
\end{equation}
If the integral kernel defined by
\begin{equation}
K^{\rm bi}(x, x')
=\int_{\Gamma} \psi(x, \gamma) \overline{\varphi(x', \gamma)} \nu(d \gamma),
\quad x, x' \in S,
\label{eqn:K_bi}
\end{equation}
is of finite rank, we can construct a finite DPP on $S$
whose correlation kernel is given by (\ref{eqn:K_bi})
following a standard method of random matrix theory
(see, for instance, Appendix C in [31]).
By the biorthonormality (\ref{eqn:bi_ortho}),
it is easy to verify that
$K^{\rm bi}$ is a projection kernel,
but it is not necessarily an orthogonal projection.
This observation means that such a DPP
is not constructed by the method reported in this paper.
Generalization of the present framework in order to cover
such DPPs associated with biorthonormal systems
is required.
Moreover, the dynamical extensions of DPPs
called {\it determinantal processes}
(see, for instance, [30])
shall be studied in the context of the present paper.

For finite DPPs, we can readily derive the
systems of {\it stochastic interacting particle systems}
whose stationary states are given by the DPPs.
For example, with $N \in \N$,
the system of stochastic differential equations (SDEs)
on $\S^1$ [26, 29],
\begin{equation}
dX_j(t)=dB_j(t)+\frac{1}{2} \sum_{\substack{1 \leq k \leq N, \cr k \not=j}}
\cot \frac{X_j(t)-X_k(t)}{2} dt,
\quad j=1, \dots, N, \quad t \geq 0,
\label{eqn:SDE1}
\end{equation}
driven by independent one-dimensional standard Brownian motions
$B_j(t), j=1, \dots, N, t \geq 0$ has the DPP
$(\Xi, K^{\AN}, \lambda_{[0, 2 \pi)}(dx))$
given in Section \ref{sec:Lie_group} as a stationary probability measure.
Another example is given by
the system of SDEs on $\C$,
\begin{equation}
d Z_j(t) = dB^{\C}_j(t)
- \frac{(N+1) Z_j(t)}{1+|Z_j(t)|^2} dt
+ \sum_{\substack{1 \leq k \leq N, \cr k \not=j}}
\frac{Z_j(t)-Z_k(t)}{|Z_j(t)-Z_k(t)|^2} dt,
\quad j=1, \dots, N, \quad t \geq 0,
\label{eqn:SDE2}
\end{equation}
$N \in \N$,
driven by independent complex Brownian motions
$B^{\C}_j(t) :=B^{\rR}_j(t)+i B^{\rI}_j(t)$,
where $B^{\rR}_j(t), B^{\rI}_j(t)$ are independent
one-dimensional standard Brownian motions,
$j=1, \dots, N, t \geq 0$,
does the DPP
$(\sum_j \delta_{z_j}, K_{G_1^{-1} G_2}^{(N)}, \lambda(dz))$
of Krishnapur [37] as a stationary probability measure,
which is obtained as the stereographic projection
of $(\Xi, K_{\S^2}^{(N)}, d \sigma_2(x))$
as explained in Remark 5 in Section \ref{sec:sphere_finite}.
A general theory has been developed by Osada {\it et al.} for
{\it infinite-dimensional stochastic differential equations} (ISDEs),
some of which have infinite DPPs as invariant probability measures
[43, 44, 45, 46, 35].
We expect to obtain the {\it universal} ISDEs along the limit theorems
given in Propositions \ref{thm:limit_Euclid} and \ref{thm:S2_Ginibre}
taking account of the fact that (\ref{eqn:SDE1}) and (\ref{eqn:SDE2})
might give useful approximations to characterize
Osada's Dyson/Ginibre ISDEs.

\vskip 0.5cm
\noindent{\bf Acknowledgements} \,
One of the present authors (MK)  thanks Christian Krattenthaler very much
for his hospitality in Fakult\"{a}t f\"{u}r Mathematik, Universit\"{a}t Wien,
where this study was started
on his sabbatical leave from Chuo University.
The authors express their gratitude to Michael Schlosser,
Masatake Hirao, and Shinji Koshida for valuable discussion.
This work was supported by
the Grant-in-Aid for Scientific Research
(C) (No.26400405), (No.19K03674),
(B) (No.18H01124), and
(S) (No.16H06338)
of Japan Society for the Promotion of Science.

%%%%%%%%%% APPENDICES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
%%%  Appendix A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\SSC{Weyl Denominator Formulas}
\label{sec:Weyl_denominators}

The {\it Weyl denominator formulas} for classical root systems
play a fundamental role in Lie theory and related area.
For reduced root systems they are given in the form,
\[
\sum_{w \in W} \det(w) e^{w(\rho)-\rho}
=\prod_{\alpha \in R_{+}}(1-e^{-\alpha}),
\]
where $W$ is the Weyl group, $R_+$ the set of positive roots
and $\rho=\frac{1}{2} \sum_{\alpha \in R_+} \alpha$.

For classical root systems $\AN, \BN, \CN$ and $\DN$,
$N \in \N$,
the explicit forms are given as follows,
\begin{align}
\mbox{ (type $\AN$)}
\quad & \det_{1 \leq j, k \leq N} \Big(z_k^{j-1} \Big)
= \prod_{1 \leq j < k \leq N} (z_k-z_j),
\nonumber\\
\mbox{ (type $\BN$)}
\quad &
\det_{1 \leq, j, k \leq N}
\Big( z_k^{j-N}-z_k^{N+1-j} \Big)
= \prod_{\ell=1}^{N} z_{\ell}^{1-N} (1-z_{\ell})
\prod_{1 \leq j < k \leq N} (z_k-z_j)(1-z_j z_k),
\nonumber\\
\mbox{ (type $\CN$)}
\quad &
\det_{1 \leq, j, k \leq N}
\Big( z_k^{j-N-1}-z_k^{N+1-j} \Big)
= \prod_{\ell=1}^{N} z_{\ell}^{-N} (1-z_{\ell}^2)
\prod_{1 \leq j < k \leq N} (z_k-z_j)(1-z_j z_k),
\nonumber\\
\mbox{ (type $\DN$)}
\quad &
\det_{1 \leq, j, k \leq N}
\Big( z_k^{j-N}+z_k^{N-j} \Big)
= 2 \prod_{\ell=1}^{N} z_{\ell}^{1-N}
\prod_{1 \leq j < k \leq N} (z_k-z_j)(1-z_j z_k),
\label{eqn:Weyl_denominator}
\end{align}
respectively. See, for instance, [47].

If we change the variables as
\begin{equation}
z_k=e^{-2 i \zeta_k}, \quad \zeta_k \in \C, \quad k=1, \dots, N,
\label{eqn:x_to_z}
\end{equation}
then, the following equalities are derived from
the above.

\begin{lem}
\label{thm:Weyl_trigonometric}
For $\zeta_k \in \C, k=1, \dots,N$,
the following equalities are established.
\begin{align*}
{\rm (type} \, \AN {\rm )}
\quad &
\det_{1 \leq j, k \leq N}
\Big[ e^{- i (\cN^{\AN}-2J^{\AN}(j)) \zeta_k} \Big]
= (2i) ^{N(N-1)/2}
\prod_{1 \leq j < k \leq N}
\sin(\zeta_k-\zeta_j).
\nonumber\\
{\rm (type} \, \BN {\rm )}
\quad &
\det_{1 \leq j, k \leq N}
\Big[ \sin\{(\cN^{\BN}-2 J^{\BN}(j)) \zeta_k\} \Big]
\nonumber\\
& \qquad
= 2^{N(N-1)}
\prod_{\ell=1}^N \sin \zeta_{\ell}
\prod_{1 \leq j < k \leq N}
\sin(\zeta_k-\zeta_j) \sin(\zeta_k+\zeta_j),
\nonumber\\
{\rm (type} \, \CN {\rm )}
\quad &
\det_{1 \leq j, k \leq N}
\Big[ \sin \{ (\cN^{\CN}-2 J^{\CN}(j)) \zeta_k\} \Big]
\nonumber\\
& \qquad
= 2^{N(N-1)}
\prod_{\ell=1}^N \sin(2 z_{\ell})
\prod_{1 \leq j < k \leq N}
\sin (\zeta_k-\zeta_j) \sin (\zeta_k+\zeta_j),
\nonumber\\
{\rm (type} \, \DN {\rm )}
\quad &
\det_{1 \leq j, k \leq N}
\Big[ \cos \{(\cN^{\DN}-2 J^{\DN}(j)) \zeta_k \} \Big]
\nonumber\\
& \qquad
= 2^{(N-1)^2}
\prod_{1 \leq j < k \leq N}
\sin(\zeta_k-\zeta_j) \sin (\zeta_k+\zeta_j),
\end{align*}
where $\cN^{\RN}$ and $J^{\RN}(j)$,
$\RN=\AN, \BN, \CN, \DN$, are given by
(\ref{eqn:N_R_classic}) and (\ref{eqn:J_R_classic}).
\end{lem}

%%%  Appendix B %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\SSC{Jacobi Theta Functions}
\label{sec:Jacobi}

\[
z=e^{v \pi i}, \quad q=e^{\tau \pi i},
\]
for $v \in \C$ and $\tau \in \H$.
The Jacobi theta functions are defined as follows [56, 42],
\begin{align}
\vartheta_0(v; \tau) &=
\sum_{n \in \Z} (-1)^n q^{n^2} z^{2n}
= 1 + 2 \sum_{n=1}^{\infty} (-1)^n e^{\tau \pi i n^2} \cos (2n \pi v),
\nonumber\\
\vartheta_1(v; \tau) &= i \sum_{n \in \Z} (-1)^n q^{(n-1/2)^2} z^{2n-1}
= 2 \sum_{n=1}^{\infty} (-1)^{n-1} e^{\tau \pi i (n-1/2)^2} \sin\{ (2n-1) \pi v \},
\nonumber\\
\vartheta_2(v; \tau)
&= \sum_{n \in \Z} q^{(n-1/2)^2} z^{2n-1}
= 2 \sum_{n=1}^{\infty} e^{\tau \pi i (n-1/2)^2} \cos \{ (2n-1) \pi v\},
\nonumber\\
\vartheta_3(v; \tau)
&=\sum_{n \in \Z} q^{n^2} z^{2n}
= 1 + 2 \sum_{n=1}^{\infty} e^{\tau \pi i n^2} \cos (2n \pi v).
\label{eqn:theta}
\end{align}
(Note that the present functions
$\vartheta_{\mu}(v; \tau), \mu=1,2,3$ are denoted by
$\vartheta_{\mu}(\pi v,q)$,
and $\vartheta_0(v;\tau)$ by $\vartheta_4(\pi v,q)$ in [56].)
For $\Im \tau >0$, $\vartheta_{\mu}(v; \tau)$, $\mu=0,1,2,3$
are holomorphic for $|v| < \infty$.
The parity with respect to $v$ is given by
\begin{equation}
\vartheta_1(-v; \tau)=-\vartheta_1(v; \tau),
\quad
\vartheta_{\mu}(-v; \tau)=\vartheta_{\mu}(v; \tau), \quad
\mu=0,2,3,
\label{eqn:even_odd}
\end{equation}
and they have the quasi-double-periodicity;
\begin{align}
\vartheta_{\mu}(v+1; \tau) &=
\begin{cases}
\vartheta_{\mu}(v; \tau), & \mu=0, 3,
\cr
- \vartheta_{\mu}(v; \tau), & \mu=1, 2,
\end{cases}
\label{eqn:quasi_periodic1}
\\
\vartheta_{\mu}(v+\tau; \tau) &=
\begin{cases}
-e^{-(2v+\tau) \pi i } \vartheta_{\mu}(v; \tau),
& \mu=0, 1,
\cr
e^{-(2v+\tau) \pi i } \vartheta_{\mu}(v; \tau),
& \mu=2,3.
\end{cases}
\label{eqn:quasi_periodic2}
\end{align}
The following relations are derived by (\ref{eqn:theta}),
\begin{align}
\vartheta_0 \left( v+\frac{\tau}{2}; \tau \right)
&= i e^{-(v+\tau/4) \pi i} \vartheta_1(v; \tau),
\nonumber\\
\vartheta_1 \left( v+\frac{\tau}{2}; \tau \right)
&= i e^{-(v+\tau/4) \pi i} \vartheta_0(v; \tau),
\nonumber\\
\vartheta_2 \left( v+\frac{\tau}{2}; \tau \right)
&= e^{-(v+\tau/4) \pi i} \vartheta_3(v; \tau),
\nonumber\\
\vartheta_3 \left( v+\frac{\tau}{2}; \tau \right)
&= e^{-(v+\tau/4) \pi i} \vartheta_2(v; \tau).
\label{eqn:half_tau}
\end{align}
By the definition (\ref{eqn:theta}),
when $\tau \in \H$,
\begin{align}
& \vartheta_1(0; \tau)=\vartheta_1(1; \tau)=0, \qquad
\vartheta_1(x; \tau) > 0, \quad x \in (0,1),
\nonumber\\
& \vartheta_2(-1/2; \tau)=\vartheta_2(1/2; \tau)=0, \qquad
\vartheta_2(x; \tau) > 0, \quad x \in (-1/2, 1/2),
\nonumber\\
& \vartheta_0(x; \tau) > 0, \quad \vartheta_3(x; \tau) > 0, \quad x \in \R.
\label{eqn:theta_values}
\end{align}
The asymptotics
\begin{align}
& \vartheta_0(v; \tau) \sim 1, \quad
\vartheta_1(v; \tau) \sim 2 e^{\tau \pi i/4} \sin (\pi v), \quad
\vartheta_2(v; \tau) \sim 2 e^{\tau \pi i/4} \cos(\pi v), \quad
\vartheta_3(v; \tau) \sim 1,
\nonumber\\
&\qquad \qquad \qquad \mbox{in} \quad
\Im \tau \to + \infty \quad
(i.e., \quad q=e^{\tau \pi i} \to 0)
\label{eqn:theta_asym}
\end{align}
are known.
We will use the following functional equations known as
{\it Jacobi's imaginary transformation} [56, 42],
\begin{align}
\vartheta_0(v; \tau)
&= e^{\pi i/4} \tau^{-1/2} e^{-\pi i v^2/\tau}
\vartheta_2 \left( \frac{v}{\tau}; - \frac{1}{\tau} \right),
\nonumber\\
\vartheta_1(v; \tau)
&= e^{3 \pi i/4} \tau^{-1/2} e^{-\pi i v^2/\tau}
\vartheta_1 \left( \frac{v}{\tau}; - \frac{1}{\tau} \right),
\nonumber\\
\vartheta_2(v; \tau)
&= e^{\pi i/4} \tau^{-1/2} e^{-\pi i v^2/\tau}
\vartheta_0 \left( \frac{v}{\tau}; - \frac{1}{\tau} \right),
\nonumber\\
\vartheta_3(v; \tau)
&= e^{\pi i/4} \tau^{-1/2} e^{-\pi i v^2/\tau}
\vartheta_3 \left( \frac{v}{\tau}; - \frac{1}{\tau} \right).
\label{eqn:Jacobi_imaginary}
\end{align}

%%%  Appendix C %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\SSC{Macdonald Denominators}
\label{sec:Macdonald_denominators}

Assume that $N \in \N$.
As extensions of the Weyl denominators for
classical root systems, Rosengren and Schlosser [47]
studied the {\it Macdonald denominators} for the seven
types of irreducible reduced affine root systems [39],
$W^{\RN}(z)$, $\z=(z_1, \dots, z_N) \in \C^N$,
$\RN=\AN$, $\BN, \BNv, \CN, \CNv, \BCN, \DN$,
$N \in \N$. (See also [55, 36].)
For $\tau \in \H$,
up to trivial factors they are written using the Jacobi theta functions
as follows.
\begin{align}
W^{\AN}(\z; \tau) &=
\prod_{1 \leq j < k \leq N} \vartheta_1(z_k-z_j; \tau),
\nonumber\\
W^{\BN}(\z; \tau) &=
\prod_{\ell=1}^N \vartheta_1(z_{\ell}; \tau)
\prod_{1 \leq j < k \leq N} \Big\{
\vartheta_1(z_k-z_j; \tau) \vartheta_1(z_k+z_j; \tau) \Big\},
\nonumber\\
W^{\BNv}(\z; \tau) &=
\prod_{\ell=1}^N \vartheta_1(2 z_{\ell}; 2 \tau)
\prod_{1 \leq j < k \leq N} \Big\{
\vartheta_1(z_k-z_j; \tau) \vartheta_1(z_k+z_j; \tau) \Big\},
\nonumber\\
W^{\CN}(\z; \tau) &=
\prod_{\ell=1}^N \vartheta_1(2 z_{\ell}; \tau)
\prod_{1 \leq j < k \leq N} \Big\{
\vartheta_1(z_k-z_j; \tau) \vartheta_1(z_k+z_j; \tau) \Big\},
\nonumber\\
W^{\CNv}(\z; \tau) &=
\prod_{\ell=1}^N \vartheta_1 \left(z_{\ell}; \frac{\tau}{2} \right)
\prod_{1 \leq j < k \leq N} \Big\{
\vartheta_1(z_k-z_j; \tau) \vartheta_1(z_k+z_j; \tau) \Big\},
\nonumber\\
W^{\BCN}(\z; \tau) &=
\prod_{\ell=1}^N \Big\{ \vartheta_1(z_{\ell}; \tau)
\vartheta_0(2 z_{\ell}; 2 \tau)
\Big\}
\prod_{1 \leq j < k \leq N} \Big\{
\vartheta_1(z_k-z_j; \tau) \vartheta_1(z_k+z_j; \tau) \Big\},
\nonumber\\
W^{\DN}(\z; \tau) &=
\prod_{1 \leq j < k \leq N} \Big\{
\vartheta_1(z_k-z_j; \tau) \vartheta_1(z_k+z_j; \tau) \Big\}.
\label{eqn:Macdonald_denominators}
\end{align}

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\end{document}