Expected Riesz Energy of Some Determinantal Processes on Flat Tori

We compute the expected Riesz energy of random points on flat tori drawn from certain translation invariant determinantal processes and determine the process in the family providing the optimal asymptotic expected Riesz energy.


Introduction
Our objective is to study the asymptotics of the expected Riesz energy of certain point processes (random finite point configurations) in a flat torus ⊂ R d . If is a lattice in R d (i.e., = AZ d for some nonsingular square matrix A), we identify the fundamental domain  For the sphere S 2 , the authors in [1] estimate asymptotically the expected energy of points of the so-called spherical ensemble. In [3], the authors study the harmonic ensemble in S d and prove, in some cases, the optimality of the expected asymptotic energy of this process among rotation invariant determinantal processes. In both cases, the expected asymptotic energy was used to get upper bounds for the minimal Riesz energy. Here, we also study the optimality of the expected asymptotic energy among a collection of determinantal processes invariant under translations, and it turns out that the best process can be found as an easy consequence of Riesz's rearrangement inequality.
This provides explicit examples with the lowest energy bounds known on the torus in high dimensions.

Riesz Energy
To define the Riesz energy in this periodic setting, we follow [8,9], see also [6,Section 9]. Given a lattice = AZ d ⊂ R d , the Epstein-Hurwitz zeta function for is defined, for s > d, as and as +∞ when x ∈ . Observe that ζ (s; x) is the -periodic potential generated by the Riesz kernel |x| −s .
When s ≤ d, the sum above is infinite for all x ∈ R d . For fixed x ∈ R d \ , define the function Then, F s, (x) is an entire function of s, and therefore by the relation we obtain an analytic continuation of ζ (s; x) to s ∈ C\{d}. Observe that the function 1/ (s) is entire and that all the sums in (1) converge uniformly. We are interested in the range 0 < s < d.
For ω = (x 1 , . . . , x N ) ∈ N , define, for 0 < s < d, the periodic Riesz s-energy of ω by and the minimal periodic Riesz s-energy by
We denote by X a (simple) random point process in a compact set ⊂ R d and let μ be the normalized Lebesgue measure. A way to describe the process is to specify the random variable counting the number of points of the process in D, for all Borel sets D ⊂ . We denote this random variable by X (D).
These point processes are characterized by their joint intensity functions ρ k in k satisfying that for any family of mutually disjoint subsets D 1 , . . . , D k ⊂ . We assume that ρ k (x 1 , . . . , x k ) = 0 when x i = x j for i = j.
A random point process is called determinantal with kernel K : × → C if it is simple and the joint intensities with respect to a background measure μ are given by for every k ≥ 1 and x 1 , . . . , x k ∈ .
To define the processes, we will consider only projection kernels.

Definition 1
We say that K is a projection kernel if it is a Hermitian projection kernel; i.e., the integral operator in L 2 (μ) with kernel K is self-adjoint and has eigenvalues 1 and 0.
By Macchi-Soshnikov's theorem [4, Theorem 4.5.5], a projection kernel K (x, y) defines a determinantal process and it has N points almost surely if the trace for the corresponding integral operator equals N , i.e., if Observe that the random vector in N generated with density is a determinantal process with the right marginals; i.e., the joint intensities are given by determinants of the kernel [2, Remark 4.2.6].
Given now a function f : × → [0, ∞), it is easy to compute the expected pair potential energy [4, Formula (1.2.2)]: Proposition 1 Let K (x, y) be a projection kernel with trace N in , and let ω = (x 1 , . . . , x N ) ∈ N be N random points generated by the corresponding determinantal point process. Then, for any measurable f :

Flat Torus
In our setting, we take as ⊂ R d the flat torus R d / , for some lattice with dual * .
To construct the kernel, we consider for w ∈ * , the Laplace-Beltrami eigenfunctions f w (u) = e 2πi u,w of eigenvalue −4π 2 w, w . Then has compact support, and we define the kernels, and the corresponding determinantal point processes on the flat torus . For these processes (we are not going to distinguish between the integral operator defined by the kernel K N and the kernel itself), we get points almost surely.

Examples
is the Dirichlet kernel.

Some Known Results About Minimal Periodic Riesz s-Energy
It was shown in [9] that for 0 < s < d, there exists a constant C s,d independent of such that for N → ∞, The constant C s,d above is not known (unless d = 1). In [9] the authors found an upper bound in terms of the Epstein zeta function. Recall that for a lattice ⊂ R d , the Epstein zeta function ζ (s) defined by can be extended analytically (as in (1)) to C \ {d}. One can see easily that ζ (0) = −1 and the residue of ζ (s) in d is 2π d/2 / (d/2) = ω d−1 . The result in [9,Corollary 3] is that for 0 < s < d, where runs on the lattices with | | = 1. It has been conjectured (see [5]) that if d = 2, 4, 8, or 24, then C s,d = ζ d (s), where d denotes (respectively) the hexagonal lattice, the D 4 lattice, the E 8 lattice, and the Leech lattice (scaled to have | d | = 1). When d = 1, indeed, C s,1 = ζ Z (s) = 2ζ(s). For d = 2, it is known, due to the work of several authors, that inf ζ (s) is attained for the triangular lattice, see [11] where the result is deduced from the corresponding result for theta functions. It is observed in [12] that from Siegel's integration formula, it follows that where dλ d is the volume measure in the space of lattices [14, p. 172]. One deduces then that C s,d < 0, although for large dimensions there are no examples providing negative bounds. Indeed, from [12], see also [13,Theorem 1], all explicitly known lattices in large dimensions are such that the corresponding Epstein zeta functions have a zero in 0 < s < d; i.e., the analogue of the Riemann hypothesis fails for Epstein zeta functions, see Remark 2.

Expected Energies
By Proposition 1, the expected periodic Riesz s-energy of t N = tr(K N ) random points ω = (x 1 , . . . , x t N ) drawn from the determinantal process defined by the kernel It is easy to see, [8], that for 0 < s < d, , and therefore by translation invariance, Our first result is a nice closed expression for the integral above.

Theorem 1
Let ω = (x 1 , . . . , x t N ) be drawn from the determinantal process on the flat torus R d / given by the kernel Then, for 0 < s < d, Remark 1 Observe that for N random points chosen independently and uniformly in , the expected energy is given by Therefore, the improvement in the expected energy for the determinantal case comes from the last summand in (4). To get a good upper bound for the minimal energy, we want to maximize the sum This is not an easy task in general. For example, when t N = 2, this would lead to finding the shortest nonzero vector in the lattice * ; i.e., or equivalently, the density of the densest lattice sphere packing.
Proof To compute the integral in (3), we write we use the expression for F s, (u) in (1) (observe that the sums converge uniformly), and interchange the order of sums and integrals, getting because e 2πi u,w−w +η dμ(u) = δ w −w,η , for w, w ∈ * and η ∈ * \ {0}.
Putting everything together, and using that for s < d, This last integral converges for all s < d, and using that (for α < −1), we get the result. Now we define a way to get different invariant kernels (by choosing different sequences of functions κ), and we estimate the corresponding expected energies. Then, for 0 < s < d, if ω = (x 1 , . . . , x t N ) ∈ t N are t N points drawn from the determinantal process defined by κ D, , The measure ν is a multiple of the Lebesgue measure such that if * is a fundamental domain for * , then ν( * ) = 1.
Proof From Theorem 1, we get that It remains to show that Observe that

Now the result follows from
Indeed, it is clear that for a fixed > 0, the corresponding limit in (5) holds if we replace the Riesz kernel by the truncated (continuous) kernel i.e., To simplify notation, assume that A is the identity matrix. If z, z ∈ Z d are such that for any z in a cube of side of length N −1/d containing z . Therefore, one can easily bound the Riemann sum which is small because the integral converges. Finally, and we get the result.
A natural question is now, given a fixed lattice , to find the optimal D ⊂ R d in the definition of κ D, . By the result above, all we have to check is what is the domain giving the larger potential I D ν . The result follows from the Riesz rearrangement inequality [10, p. 87].

Theorem 2 (Riesz rearrangement inequality). Given f, g, H nonnegative functions in R d with h(x) = H (|x|) symmetrically decreasing, then
where f * , g * are the symmetric decreasing rearrangements of f and g.
We apply the result for f = g = χ D , getting f * = g * = χ D * , where D * is the open ball centered at the origin with |D * | = |D|.
is the surface measure of the unit ball S d−1 in R d , then

An Upper Bound for the Minimal Periodic Riesz s-Energy
It is clear that and therefore from the results above, we get that C s,d in (2) satisfies, for 0 < s < d, where D is any bounded domain such that |D|| | = 1. By the discussion above, the best choice is to take the set D to be a ball.  Fig.1. However, it is observed in [12] that no negative bound was known for C s,d for large dimensions. Indeed, it was proved in [13] (see also [14, 4.4.4.]) that for 0 < δ < 1 and d sufficiently large, In [9] the authors consider also the case of logarithmic interaction. This can be seen as a limiting case in the sense that (t s − 1)/s → log t when s → 0. Our approach works also in this setting, but the results are far from optimal in high dimensions, as they depend on the derivative, see Fig. 1. A similar phenomenon was observed in [3].