Optimal Discrete Riesz Energy and Discrepancy

The Riesz $s$-energy of an $N$-point configuration in the Euclidean space $\mathbb{R}^{p}$ is defined as the sum of reciprocal $s$-powers of all mutual distances in this system. In the limit $s\to0$ the Riesz $s$-potential $1/r^s$ ($r$ the Euclidean distance) governing the point interaction is replaced with the logarithmic potential $\log(1/r)$. In particular, we present a conjecture for the leading term of the asymptotic expansion of the optimal \IL_2-discrepancy with respect to spherical caps on the unit sphere in $\mathbb{R}^{d+1}$ which follows from Stolarsky's invariance principle [Proc. Amer. Math. Soc. 41 (1973)] and the fundamental conjecture for the first two terms of the asymptotic expansion of the optimal Riesz $s$-energy of $N$ points as $N \to \infty$.Comment: 8 page

Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres

We study expected Riesz s-energies and linear statistics of some determinantal processes on the sphere $\mathbb{S}^{d}$. In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble on $\mathbb{S}^{d}$. With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere. We prove a comparison result for Riesz 2-energies of points defined through determinantal point processes associated with isotropic kernels. As a corollary we get that the Riesz 2-energy of the harmonic ensemble is optimal among ensembles defined by isotropic kernels with the same trace. Finally, we study the variance of smooth and rough linear statistics for the harmonic ensemble and compare the results with the variance for the spherical ensemble (in $\mathbb{S}^{d}$)

"Magic" numbers in Smale's 7th problem

Smale's 7-th problem concerns N-point configurations on the 2-dim sphere which minimize the logarithmic pair-energy V_0(r) = -ln r averaged over the pairs in a configuration; here, r is the chordal distance between the points forming a pair. More generally, V_0(r) may be replaced by the standardized Riesz pair-energy V_s(r)= (r^{-s} -1)/s, which becomes - ln r in the limit s to 0, and the sphere may be replaced by other compact manifolds. This paper inquires into the concavity of the map from the integers N>1 into the minimal average standardized Riesz pair-energies v_s(N) of the N-point configurations on the 2-sphere for various real s. It is known that v_s(N) is strictly increasing for each real s, and for s<2 also bounded above, hence "overall concave." It is (easily) proved that v_{-2}(N) is even locally strictly concave, and that so is v_s(2n) for s<-2. By analyzing computer-experimental data of putatively minimal average Riesz pair-energies v_s^x(N) for s in {-1,0,1,2,3} and N in {2,...,200}, it is found that {v}_{-1}^x(N) is locally strictly concave, while v_s^x(N) is not always locally strictly concave for s in {0,1,2,3}: concavity defects occur whenever N in C^{x}_+(s) (an s-specific empirical set of integers). It is found that the empirical map C^{x}_+(s), with s in {-2,-1,0,1,2,3}, is set-theoretically increasing; moreover, the percentage of odd numbers in C^{x}_+(s), s in {0,1,2,3}, is found to increase with s. The integers in C^{x}_+(0) are few and far between, forming a curious sequence of numbers, reminiscent of the "magic numbers" in nuclear physics. It is conjectured that the "magic numbers" in Smale's 7-th problem are associated with optimally symmetric optimal-energy configurations.Comment: 109 pages, of which 30 are numerical data tables. Thoroughly revised version, to appear in J. Stat. Phys. under the different title: `Optimal N point configurations on the sphere: "Magic" numbers and Smale's 7th problem