发表机构
Università Unipegaso; Università degli Studi di Bergamo（佩加索大学; 贝加莫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文定义球面上的 Pearl 系综随机点集，精确计算其对数能量至 $O(\sqrt{N}\log N)$ 阶，并证明在已知显式构造中，其线性项系数可接近最小值 $(1-\log 3)/2$。
AI 中文摘要
我们定义了球面 $\mathbb S^2$ 上的一族随机点集，即 Pearl 系综，它依赖于若干参数。其对数能量的期望值可以精确计算到 $O(\sqrt{N}\log N)$ 阶项，其中 $N$ 为点数。对于适当选择的参数值，对数能量展开式中线性项的系数可以任意接近值 $(1-\log 3)/2\sim -0.0493061\ldots$。在我们目前已知的、已严格确定对数能量展开式至线性项的显式构造中，Pearl 系综给出了最小的系数。
英文摘要
We define a family of random sets of points on the sphere $\mathbb S^2$, the Pearl ensemble, depending on several parameters. The expected value of the logarithmic energy can be computed exactly up to terms of order $O(\sqrt{N}\log N)$, with $N$ the number of points. For properly chosen values of the parameters, the coefficient of the linear term in the expansion of the logarithmic energy can be taken as close as desired to the value $(1-\log 3)/2\sim -0.0493061\ldots$. Among the explicit constructions currently known to us for which the logarithmic energy expansion has been rigorously determined up to the linear term, the Pearl ensemble yields the smallest coefficient.
Comments24 pages, 1 figure
