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Register a new userMinimal Riesz energy on the sphere for axis-supported external fields
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We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S^d in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x-y|^(-s) with d-2 <= s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on S^d is determined. The special case s = d-2 yields interesting phenomena, which we investigate in detail. A weak* asymptotic analysis is provided as s goes to (d-2)^+.
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We survey known results and present estimates and conjectures for the next-order term in the asymptotics of the optimal logarithmic energy and Riesz $s$-energy of $N$ points on the unit sphere in $mathbb{R}^{d+1}$, $dgeq 1$. The conjectures are based on analytic continuation assumptions (with respect to $s$) for the coefficients in the asymptotic expansion (as $Nto infty$) of the optimal $s$-energy.
Let $Lambda$ be a lattice in ${bf R}^d$ with positive co-volume. Among $Lambda$-periodic $N$-point configurations, we consider the minimal renormalized Riesz $s$-energy $mathcal{E}_{s,Lambda}(N)$. While the dominant term in the asymptotic expansion of $mathcal{E}_{s,Lambda}(N)$ as $N$ goes to infinity in the long range case that $0<s<d$ (or $s=log$) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for $s>0$ they are of the form $C_{s,d}|Lambda|^{-s/d}N^{1+s/d}$ and $-frac{2}{d}Nlog N+left(C_{log,d}-2zeta_{Lambda}(0)right)N$ where we show that the constant $C_{s,d}$ is independent of the lattice $Lambda$.
For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a large class of manifolds A having finite, positive d-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the d-dimensional sphere.
For the unit sphere S^d in Euclidean space R^(d+1), we show that for d-1<s<d and any N>1, discrete N-point minimal Riesz s-energy configurations are well separated in the sense that the minimal distance between any pair of distinct points in such a configuration is bounded below by C/N^(1/d), where C is a positive constant depending on s and d.
Let $A$ be a compact $d$-rectifiable set embedded in Euclidean space $RR^p$, $dle p$. For a given continuous distribution $sigma(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, our earlier results provided a method for generating $N$-point configurations on $A$ that have asymptotic distribution $sigma (x)$ as $Nto infty$; moreover such configurations are quasi-uniform in the sense that the ratio of the covering radius to the separation distance is bounded independent of $N$. The method is based upon minimizing the energy of $N$ particles constrained to $A$ interacting via a weighted power law potential $w(x,y)|x-y|^{-s}$, where $s>d$ is a fixed parameter and $w(x,y)=left(sigma(x)sigma(y)right)^{-({s}/{2d})}$. Here we show that one can generate points on $A$ with the above mentioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most $r_N=C_N N^{-1/d}$ from each other, with $C_N$ being a positive sequence tending to infinity arbitrarily slowly. To do this we minimize the energy with respect to a varying truncated weight $v_N(x,y)=Phi(left|x-yright|/r_N)w(x,y)$, where $Phi:(0,infty)to [0,infty)$ is a bounded function with $Phi(t)=0$, $tgeq 1$, and $lim_{tto 0^+}Phi(t)=1$. This reduces, under appropriate assumptions, the complexity of generating $N$ point `low energy discretizations to order $N C_N^d$ computations.
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