Asymptotics of greedy energy points

Abstract:

For a symmetric kernel $k:X\times X \rightarrow \mathbb {R}\cup \{+\infty \}$ on a locally compact metric 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 $N\rightarrow \infty$ that is asymptotically the same as $\mathcal {E}(A,N):=\min \{\sum _{i\neq j}k(x_{i},x_{j}):x_{1},\ldots ,x_{N}\in A\}$), 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 (the case $s=\infty$), and for weighted Riesz kernels. For greedy best-packing points we provide a simple counterexample to a conjecture attributed to L. Bos.