Riesz energy, L2$L^2$ discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus

Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so‐called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere Sd$\mathbb {S}^d$ and the flat torus Td$\mathbb {T}^d$ , and the so‐called spherical ensemble on S2$\mathbb {S}^2$ , which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega‐Cerdà on the Riesz s$s$ ‐energy of the harmonic ensemble to the nonsingular regime s<0$s<0$ , and as a corollary find the expected value of the spherical cap L2$L^2$ discrepancy via the Stolarsky invariance principle. We find the expected value of the L2$L^2$ discrepancy with respect to axis‐parallel boxes and Euclidean balls of the harmonic ensemble on Td$\mathbb {T}^d$ . We also show that the spherical ensemble and the harmonic ensemble on S2$\mathbb {S}^2$ and T2$\mathbb {T}^2$ with N$N$ points attain the optimal rate N−1/2$N^{-1/2}$ in expectation in the Wasserstein metric W2$W_2$ , in contrast to independent and identically distributed random points, which are known to lose a factor of (logN)1/2$(\log N)^{1/2}$ .


INTRODUCTION
Point processes with built-in repulsion are classical models for fermionic particle systems in quantum mechanics.A similar phenomenon of repulsion occurs for the eigenvalues of random matrices and the zeros of random polynomials (see, e.g., [28, chapter 4]).Many repulsive systems fall into the category of determinantal point processes: these are simple point processes whose joint intensities are given by the determinant of a suitable positive semidefinite kernel.Such systems provide a natural way of distributing points on a compact manifold, with the built-in repulsion ensuring a more even point distribution than, say, independent and identically distributed random variables.In Section 2, we give the precise definitions of all notions appearing here in the Introduction.
The main goal of this paper is to demonstrate the repulsive properties of the so-called harmonic ensemble on the unit sphere   and the flat torus   , and the so-called spherical ensemble on  2 .In particular, these systems are closer to uniformity in terms of the nonsingular Riesz energy, in  2 discrepancy and in the Wasserstein metric  2 than uniformly distributed independent and identically distributed points.
The harmonic ensemble on a compact manifold  is the determinantal point process defined by the reproducing kernel of a finite-dimensional subspace of  2 (), typically that spanned by the Laplace eigenfunctions corresponding to eigenvalues up to a certain threshold.It was first introduced by Beltrán, Marzo, and Ortega-Cerdà on the sphere   in [14], where they established the precise asymptotics of its expected singular Riesz and logarithmic energies, and studied its separation distance and linear statistics.The energy results have since been extended to projective spaces [5] and to SO(3) [13], where the Green energy was also found.The harmonic ensemble has also been considered on the flat torus in [39], where its expected periodic Riesz energy for 0 <  <  was investigated.The harmonic ensemble on both the sphere [20] and the torus [47] has been shown to be hyperuniform.
The spherical ensemble has its origins in random matrix theory.Krishnapur [33,34] showed that the complex eigenvalues of  −1 , where  and  are independent  ×  matrices with independent and identically distributed standard complex Gaussian entries, form a determinantal point process on ℂ with respect to a certain background measure.The spherical ensemble is then obtained by the stereographic projection of these eigenvalues onto the sphere  2 , and turns out to be a determinantal point process itself.Its precise distribution properties were investigated in [3], in particular both the singular and nonsingular Riesz energy and the  2 discrepancy were shown to be of optimal order among all point sets of the same size.The spherical ensemble was generalized to complex projective spaces and even-dimensional spheres in [11,12].
For the empirical spectral measure of further random matrix models, we refer to the monograph [7].In particular, estimating the convergence rate in the Kolmogorov and the Wasserstein metric on the circle (for random unitary matrices), on the real line (for random Hermitian matrices) and on the complex plane (for random non-Hermitian matrices) is a classical problem.The latter metric originates in the theory of optimal transport, and has become a powerful tool in geometric analysis, see [49] for an introduction.We follow the approach of Götze and Jalowy [24,31] to deduce sharp rates in the Wasserstein metric by using a suitable smoothing inequality.
The main contributions of the present paper, discussed in more detail in Section 2, are the following.
• We find the expected value of the nonsingular Riesz energy of the harmonic ensemble on   (Theorem 1).
• We find the expected value of the  2 discrepancy of the harmonic ensemble on   via the Stolarsky invariance principle (Corollary 2), and show that it has optimal order up to a factor of (log ) 1∕2 among all point sets of the same size.• We find the expected value of the  2 discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on   via Fourier analysis (Theorems 4 and 5).The  2 discrepancy with respect to Euclidean balls has optimal order up to a factor of (log ) 1∕2 among all point sets of the same size.• We show that the harmonic ensemble on  2 and  2 , and the spherical ensemble on  2 have distance ( −1∕2 ) from uniformity in expectation in the Wasserstein metric  2 , which is optimal among all point sets of the same size (Theorem 6).
The proof of all our results rely on the explicit formulae for the Laplace eigenfunctions on   and   .We mention without further details that similarly to [5], our approach could be extended to other two-point homogeneous spaces, on which the Laplace eigenfunctions are also explicitly known.We only comment on the complex projective space ℂℙ  , in particular on the expected value of the  2 discrepancy of the projective ensemble (Corollary 3).

Determinantal point processes
where convergence is absolute and uniform.This immediately implies that meaning   is trace class.A simple point process   on  is called a determinantal point process with kernel  if the joint intensities (correlation functions) with respect to Vol are   ( 1 ,  2 , … ,   ) = det((  ,   ))   ,=1 ,  = 1, 2, ….For every eigenfunction   , the process produces a point with probability   , which implies that the expected number of points is the trace of   .If  1 = ⋯ =   = 1, and   = 0 for all  > , then   is a projection to the subspace spanned by  1 , … ,   .In this case, we call  a projection kernel, and the corresponding determinantal point process samples exactly  points,   = { 1 , … ,   }.For the rest of the paper, we will only deal with projection kernels.In particular, for any Borel measurable function  ∶  → ℂ, and provided that the integrals exist.For the general theory of determinantal point processes, we refer to [29].
In this paper, we will work on the standard Euclidean unit sphere  =   ⊂ ℝ +1 and the flat torus  =   = ℝ  ∕ℤ  .The volumes are normalized as ) and Vol(  ) = 1.Recall the orthogonal decomposition  2 (  , Vol) = ⊕ ∞ =0   , where   is the space of spherical harmonics of degree .
) be an (arbitrary) orthonormal basis in   .We refer to [46] for a general overview of spherical harmonics.

Definition 1.
(i) Let  ∈ ℕ.The harmonic ensemble on   is the determinantal point process  = (, (ii) Let  ∈ ℕ.The harmonic ensemble on   is the determinantal point process Note that (, ) on   , respectively,   , is the reproducing kernel of the finite-dimensional subspace ⊕  =0   ⩽  2 (  , Vol), respectively, Span { 2⟨,⟩ ∶ ‖‖ ∞ ⩽ } ⩽  2 (  , Vol).The kernel is rotation invariant on   , respectively, translation invariant on   .By the above discussion, the number of points in the harmonic ensemble is deterministic, and the number of points is given by Remark 1.More generally, we could define the harmonic ensemble on a compact manifold  as the determinantal point process  = { 1 ,  2 , … ,   } with kernel  ∶  ×  → ℂ, (, ) = ∑ −1 =0   ()  (), where   ,  ⩾ 0 is an orthonormal basis in  2 (, Vol) that consists of eigenfunctions of the Laplace-Beltrami operator on , in increasing order of the eigenvalues.
Definition 2. Let  ∈ ℕ, and consider two independent  ×  matrices ,  whose entries are independent and identically distributed mean zero Gaussian random variables with density  −1  −|| 2 on ℂ.The spherical ensemble on  2 is the stereographic projection of the  complex eigenvalues of the matrix  −1  to  2 .

Riesz energy on the sphere
The continuous Riesz -energy of a Borel probability measure  on   is defined as where  0 () = Γ ′ ()∕Γ() is the digamma function.
The discrete Riesz -energy of a finite point set   = { 1 ,  2 , … ,   } ⊂   is similarly defined as Note that in the nonsingular regime  < 0,   (  ) is  2 times the continuous Riesz -energy of the corresponding empirical measure  =  −1 ∑  =1    .The optimization of the discrete and continuous Riesz energies on the sphere has been wellstudied [18].Björck [16] showed that • if −2 <  < 0, then   () is uniquely maximized by the normalized volume measure; • if  = −2, then the maximizers of   () are exactly the measures with center of mass at the origin, that is, ∫    () = 0; • if  < −2, then the maximizers of   () are exactly the measures of the form 1 2 (  +  − ) for any  ∈   .This immediately yields maximizers of the discrete Riesz energies   (  ) for  ⩽ −2.In the case  = −2, any point set with center of mass at the origin is optimal, for example, evenly spaced points on some great circle.In the case when  < −2 and  is even, taking ∕2 points at a pole and the rest at the opposite pole is optimal.
In the case −2 <  < 0, we have (see [21, with some positive constants  , ,  ′ , .The optimal point configurations   are not known in general, however, it is known that they are uniformly distributed as  → ∞ [18,Theorem 6.1.7]. Consider now the Riesz -energy   () of the harmonic ensemble  = (,   ) = { 1 ,  2 , … ,   }.Beltrán, Marzo, and Ortega-Cerdà [14] showed that its expected value is finite if and only if  <  + 2, and gave an explicit formula in the singular regime 0 ⩽  ⩽ .The fact that the expectation of the Riesz energy under such determinantal point processes converges for  <  + 2 (as opposed to  <  for independent and identically distributed points) can be explained by the vanishing to the second order of the joint intensity  2 (, ) for  = .
In particular, for 0 <  <  they showed that ) Γ ) . (5) In this paper, we extend their result to the nonsingular regime  < 0.

2.3
discrepancy on the sphere The sets are called spherical caps in   .The spherical cap  2 discrepancy of a finite point set .
Stolarsky's original proof in [48] has been simplified in [15,19], and the result has been generalized to projective spaces by Skriganov [45].We refer the reader to [42,44] for a detailed study of the   discrepancy on compact metric spaces, and to [23] for a general exposition on discrepancy.Theorem 1 with  = −1 and the Stolarsky invariance principle thus immediately imply that the expected value of the spherical cap  2 discrepancy of the harmonic ensemble has optimal order up to a factor of (log ) 1∕2 .
In comparison, for  uniformly distributed independent and identically distributed points  = { 1 ,  2 , … ,   } on   , we have

𝑳 𝟐 discrepancy on the complex projective space
We can similarly define the  2 discrepancy of a finite point set , The  2 discrepancy on ℂℙ  was studied in [43,45], and the Riesz energy in [5,11,22].The optimal order of the former for any set of  points is [43, Theorem 2.2] As a generalization of the spherical ensemble, Beltrán and Etayo [11] introduced the socalled projective ensemble on ℂℙ  .Given  ∈ ℕ, the projective ensemble  = (, ℂℙ  ) = { 1 ,  2 , … ,   } has  = ( +  ) points and projection kernel satisfying with ,  ∈ ℂℙ  being identified with corresponding points in ℂ +1 , for ease of notation.The expected Riesz -energy of the projective ensemble was shown in [11] to satisfy for all †  < 2,  ≠ 0, The previous formula with  = −1 combined with the invariance principle ( 6) immediately implies that the  2 discrepancy of the projective ensemble attains the optimal rate in expectation.

Corollary 3. The projective ensemble
In the case  = 1, ℂℙ 1 corresponds to a 2-dimensional sphere of radius 1∕2, and the projective ensemble reduces to the spherical ensemble.
Finally, we note that the construction of the projective ensemble involved first constructing a projection kernel on ℂ  from an orthogonal system (of monomials with weight functions), then applying an inverse stereographic map from ℂ  to ℂℙ  to achieve the kernel on the projective space.It is possible to construct determinantal point processes on the real, complex, quaternion, and octonian projective spaces from an orthogonal system of functions in  2 (  , ℂ) in a similar manner.However, monomials are generally not orthogonal on ℝ  , and are not complexvalued on ℍ  and   , so the projective ensemble cannot be generalized to the corresponding projective spaces.

𝑳 𝟐 discrepancies on the torus
We consider two types of  2 discrepancies on the torus   .The periodic  2 discrepancy is defined in terms of axis-parallel boxes, whereas the ball  2 discrepancy is in terms of Euclidean balls.
For ,  ∈ , the "periodic interval" [, ] per is the arc on the circle connecting the points  and  counterclockwise.This extends naturally to higher dimensions: for ,  ∈   , the "periodic axis-parallel box" is .
Several variants of this notion have been considered before on the unit cube: the (ordinary)  2 discrepancy is defined using only axis-parallel boxes anchored at the origin, whereas the extremal  2 discrepancy uses axis-parallel boxes in the cube.For our purposes, the periodic version is the most natural, as it is compatible with the group operation on   , that is, it is invariant under translations of the point set   .In fact,  per,2 is, up to normalizing constants, identical to the diaphony of Zinterhof [52].We refer to [27] for a survey on the relationship between various versions of the  2 discrepancy and the diaphony.
The smallest possible order for any set of  points in   is (see [27, Corollary 2] and [37, Main Theorem]) per,2 (  ).
For  = 1, the lower estimate is achieved by the  th roots of unity, and for  ⩾ 2, the lower estimate is achieved for infinitely many  [27,35,41].
As an analogue of Corollary 2, we find the expected value of the periodic  2 discrepancy of the harmonic ensemble on the torus.Throughout,  denotes Euler's constant.with an implied constant depending only on .In particular, Let us now consider the  2 discrepancy with respect to balls.The geodesic metric on the flat torus  , where ‖‖ = min ∈ℤ | − |,  ∈ ℝ, denotes the distance to the nearest integer function.Let (, ) denote the open ball in the geodesic metric centered at  ∈   with radius  > 0. We define the ball  2 discrepancy of a finite point set .
Note that this is also invariant under translations of the point set   .The smallest possible order for any set of  points in   is (see [9,10,40]) We also find the expected value of the ball  2 discrepancy of the harmonic ensemble on   .
In comparison, for  uniformly distributed independent and identically distributed points  = { 1 ,  2 , … ,   } on   , we have The harmonic ensemble on   thus achieves the same rate in expectation in the periodic and the ball  2 discrepancy, and this rate is better than that for independent and identically distributed random points.For the ball  2 discrepancy, this rate is almost optimal among all point sets of the same size, as was also the case for the spherical cap  2 discrepancy.For the more sensitive periodic  2 discrepancy, however, the attained rate is far from optimal.

Wasserstein metric
Let  denote the geodesic distance on , let () be the set of Borel probability measures on , and let 1 ⩽  < ∞.The -Wasserstein metric is defined as where Coup(, ) is the set of couplings (transport plans) of  and , that is, the set of  ∈ ( × ) such that ( × ) = () and ( × ) = () for all Borel sets  ⊆ .See [49] for basic properties of   .Approximating a given probability measure by a finitely supported one is known as the quantization problem.The best rate for the normalized Riemannian volume measure is where the infimum is over all  ∈ () supported on at most  points with arbitrary weights.The constant (, ) > 0 depends only on  and the dimension , but not on the geometry of .√ , however, the precise value of (, ) is not known in general.We refer to [25,30,32] for more details.
For simplicity of notation, we will identify a finite set   = { 1 ,  2 , … ,   } ⊂  with its empirical measure, and write   (  , ) = ) . The main message of the following result is that the empirical measure of the harmonic and spherical ensembles on the 2-dimensional spaces  2 and  2 attain the optimal rate  −1∕2 in (7) in the Wasserstein metric  2 .In particular, the following estimates are sharp up to a constant factor.The appearance of the extra factor √ log  was first observed by Ajtai, Komlós, and Tusnády in connection to the optimal matching problem [1].
The optimal transport problem is much simpler on the 1-dimensional torus .In this case, we simply have  2 (  , Vol) = 2 −1∕2  per,2 (  ) for any finite set   , see Subsection 5.4.This leads to the following explicit formula for the harmonic ensemble.
In particular, .
In dimensions  ⩾ 3, uniformly distributed independent and identically distributed points  = { 1 ,  2 , … ,   } in fact match the optimum in (7): for all large enough , Following the proof of this fact in [17], the same can be deduced for the harmonic ensemble on   and   ,  ⩾ 3.More generally, the same holds for the harmonic ensemble on a compact manifold  of dimension  ⩾ 3, as defined in Remark 1, provided that (, ) is constant.
Observe that and also the rough estimate uniformly in 0 ⩽  ⩽ .
The integral in the previous formula was evaluated as the value of a hypergeometric function at the point −1 Here and   ∕( + ) =   ∕2 + (( − ) −1 ), hence The infinite series in the previous formula was evaluated using the asymptotics of the partial sums of zero-balanced hypergeometric series [38, p. 110].Further, + ( −1 log ).
Using  = (1 + ( −1 ))2  ∕!, we thus have In either case, we obtain and the claim for  < −1 follows.The infinite series in the previous formula was evaluated by a formula of Gauss on the value of hypergeometric functions at the point 1 [6, p. 66].□

4
DISCREPANCIES ON
In contrast, the variance of the exponential sum for the harmonic ensemble is the following.
The contribution of the second term in ( 14) is negligible: All other terms in ( 14) are easily seen to have an even smaller contribution.The total contribution of all  ∈ [− 1  , 1  ]  ,  ≠ 0 in ( 13) is thus .
Note that for all other terms .
For any given 1 ⩽  ⩽ , the same holds for the terms with ]  with at least two coordinates being >  1  in absolute value, is similarly seen to be ( −1− 2  ).The total contribution of all  ∈ ℤ  ∖[− 1  , 1  ]  in ( 13) is thus .

Ball 𝑳 𝟐 discrepancy
In this section, we estimate the ball  2 discrepancy of the harmonic ensemble on   , and prove Theorem 5. Our starting point is a Fourier analytic formula for the ball  2 discrepancy, similar to the diaphony (11).
Proof.Using the formula for the Fourier transform of the indicator function of the Euclidean unit ball [46, p. 155], we immediately deduce that for any 0 <  ⩽ 1∕2, the Fourier coefficients of the indicator function  (0,) of the ball (0, ) ⊂   are where   are the Bessel functions of the first kind.As  (,) () =  (0,) ( − ), for any fixed  ∈   we obtain the Fourier series expansion in the variable   (,) () = Vol((0, )) + ∑ Setting  =   and then summing over 1 ⩽  ⩽  leads to the Fourier series expansion An application of the Parseval formula thus yields the desired formula with The asymptotics of the Bessel functions [6, p. 209] shows that Hence, and the claim follows.□ For uniformly distributed independent and identically distributed points  = { 1 ,  2 , … ,   }, Proof of Theorem 5. Recall that  = (2 + 1)  .Lemmas 8 and 9 immediately yield with some positive real numbers   = 1∕(2   2 || +1 ) + (1∕|| +2 ).First note that the contribution of the terms || >  1∕ in ( 15) is negligible: Consider now the terms 0 < || ⩽  1∕ in (15).For all such terms, |  | ⩽ 2 + 1 for all 1 ⩽  ⩽ , and formula ( 14) holds.The main contribution comes from the first term on the right-hand side of ( 14): The previous sum is easily estimated in dimension  = 1.We now show how to estimate it in dimensions  ⩾ 2. Fix  1 ≠ 0, and let us estimate the inner sum over  2 , … ,   with the corre- Note that and that Integrating over the unit cube centered at the lattice point ( 2 , … ,   ) and then summing over ( 2 , … ,   ) thus leads to Vol( −2 ) −2  + ) .
Replacing the upper limit of integration in the previous integral by infinity yields ) .
Note that the previous formula holds in dimension  = 1 as well.Formula ( 17) thus simplifies to ) .
The second term on the right-hand side of ( 14) is negligible: A similar argument shows that the contribution of all later terms on the right-hand side of ( 14) is also ≪ 1∕ 1+ 1  .Hence, , which together with ( 15) and ( 16) prove the claim.□
Lemma 10.For any  > 0, Proof.By the addition formula (20), the kernel is (, ) = ∑  =0 2+1 4   (⟨, ⟩).In particular, it is constant on the diagonal: (, ) = ∕(4).An application of (2) with () =    () thus gives The fact that the Legendre polynomials form an orthogonal basis in  2 ([−1, 1]) means in particular, that the polynomial   1 ()  () can be expanded as a finite sum and consequently As a manifestation of the repulsive nature of the system in Fourier space, the main term  in the variance cancels, and we obtain Set g = ( 1 +  2 + )∕2.The remaining integral vanishes unless g ⩾  1 ,  2 , , as can be seen from comparing the degree of the product of two polynomials to the third one.Otherwise, we have the explicit formula [6, p. 321] Assume first that  ⩽ ∕2.The central binomial coefficients satisfy the general inequalities for all  ⩾ 0. Therefore, which in turn leads to .
If  > ∕2, the proof is much simpler.In this case, formula (21) yields the trivial upper bound (cf.uniformly distributed independent and identically distributed points) which suffices for the claim of the lemma.□ Proof of Theorem 6 (i).An application of the smoothing inequality (18), the triangle inequality for the  2 norm and Lemma 10 yield that for any real  > 0, and 1∕4  .
The optimal choice is  = An application of the smoothing inequality (19), the triangle inequality for the  2 norm and (24) yield that for any real  > 0, √  2 2 (, Vol) ⩽ 2 1∕2  1∕2 + 2 Let () = |{ ∈ ℤ 2 ∶ 0 < || 2 ⩽ }| denote the number of nonzero lattice points in the closed disk centered at the origin of radius  1∕2 .Let us draw a unit square centered at each of these lattice points.As these pairwise disjoint squares are all a subset of the disk of radius  1∕2 + 2 ) .
The optimal choice is  = 1∕(4 1∕3 ), and we finally obtain We emphasize that this formula holds only in dimension  = 1, and only for the distance from the Riemannian volume.In light of formula (11), the distance from uniformity in  2 , the periodic  2 -discrepancy and the diaphony are thus identical up to normalizing constants.In particular,  2 (  , Vol) = 2 −1∕2  per,2 (  ).
Proof of Theorem 7. The claim now immediately follows from formula (12).□

A C K N O W L E D G M E N T S
Bence Borda was supported by the Austrian Science Fund (FWF) project M 3260-N.Peter Grabner and Ryan Matzke were supported by the Austrian Science Fund FWF project F5503 part of the Special Research Program (SFB) "Quasi-Monte Carlo Methods: Theory and Applications".Ryan Matzke was also supported by the National Science Foundation Postdoctoral Fellowship Grant 2202877.This work was finished during a visit of Peter Grabner to Vanderbilt University, which was funded by Doug Hardin.He would like to extend his gratitude for the hospitality.The authors would also like to thank Carlos Beltrán for his helpful communications, as well as the two anonymous referees for their helpful comments and suggestions.

J O U R N A L I N F O R M AT I O N
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