Asymptotics of Greedy Energy Points

Abstract

For a symmetric kernel $k:X\times X \to \mathbb{R}\cup\{+\infty\}$ on a locally compact Hausdorff space $X$, we investigate the asymptotic behavior of greedy $k$-energy points $\{a_{i}\}_{1}^{\infty}$ for a compact subset $A\subset X$ that are defined inductively by selecting $a_{1}\in A$ arbitrarily and $a_{n+1}$ so that $\sum_{i=1}^{n}k(a_{n+1},a_{i})=\inf_{x\in A}\sum_{i=1}^{n}k(x,a_{i})$. We give sufficient conditions under which these points (also known as Leja points) are asymptotically energy minimizing (i.e. have energy $\sum_{i\neq j}^{N}k(a_{i},a_{j})$ as $N\to\infty$ that is asymptotically the same as $\mathcal{E}(A,N):=\min\{\sum_{i\neq j}k(x_{i},x_{j}):x_{1},...,x_{N}\in A\}$), and have asymptotic distribution equal to the equilibrium measure for $A$. For the case of Riesz kernels $k_{s}(x,y):=|x-y|^{-s}$, $s>0$, we show that if $A$ is a rectifiable Jordan arc or closed curve in $\mathbb{R}^{p}$ and $s>1$, then greedy $k_{s}$-energy points are not asymptotically energy minimizing, in contrast to the case $s<1$. (In fact we show that no sequence of points can be asymptotically energy minimizing for $s>1$.) Additional results are obtained for greedy $k_{s}$-energy points on a sphere, for greedy best-packing points, and for weighted Riesz kernels.