指数能量的精确构型与碰撞尺度
Exact Configurations and Collision Scales for Exponential Energy
研究概述
在一个区间上放置允许重复的点，使各点对距离 d 对应的 exp(-d) 之和最小。论文确定了全部极小位置及每个端点的精确重复点数。在单位区间上，这证明了 Emmerich 在 arXiv 2603.28179v2 第 4 节报告的经验性三分之一规律。论文还在点数和区间长度固定时，回答附录 A.3 末尾关于邻近 q 值的问题。对于 exp(-d^q)，当 q 从下方趋于 1 时，簇内移动点以不同的指数尺度靠近端点；在特殊长度处，最外层尺度转而带有对数修正。当 q 从上方趋于 1 时，端点重数保持不变，包括这些特殊长度。有限尺寸能量修正和各指数尺度共有的领先因子也在同一篇中得到证明。猜想 A.1 和 A.2 中的全局转变不属于本文的局部结果。
原文摘要（英文）
We determine the minimum of the exponential pair energy among all ordered n-point configurations on a compact interval, allowing repeated points. The unique minimizer consists of equal endpoint clusters and an equally spaced interior block. A scalar equation specifies every position, and explicit interval-length thresholds determine the cluster multiplicity. On the unit interval this proves the empirical one-third law reported by Emmerich. We also determine the first finite-size corrections to the minimum energy. For the kernels exp(-d^q) as q increases to one, each moving point in an endpoint cluster approaches the boundary on a distinct exponential scale. At a length threshold the outermost point instead has an explicit (1-q) log(1/(1-q)) scale. From above, the endpoint multiplicities persist for every fixed interval length, including the thresholds. These results describe both the exact critical configuration and its two different local perturbation regimes.
原始公开问题
Michael T. M. Emmerich: Critical phase transitions in minimum-energy configurations for the exponential kernel family e^{-|x-y|^q} on the unit interval — Section 4 empirical one-third law and the final nearby-q perturbative question in Appendix A.3, arXiv v2. This is not a solution of Conjectures A.1/A.2 or the full large-n phase diagram.
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