论文标题
框架电势的相转换] {$ p^{th} $框架电势的最小值的相变
Phase transitions for frame potentials]{Phase transitions for the minimizers of the $p^{th}$ frame potentials in $\mathbb{R}^2$
论文作者
论文摘要
给定的$ n $点$ x = \ {x_k \} _ {k = 1}^n $在$ \ mathbb {r}^2 $和一个数字$ 0 \ leq p \ leq p \ leq \ leq \ indty $中，我们调查了功能$ \ sum_ \ sum_ {k，el e el = 1}^n | x_ \ ell \ rangle |^p $。众所周知，这些最小化器中的每一个都是$ \ mathbb {r}^2 $的一个跨度集，但对它们的数字却是$ p $和$ n $的函数，尤其是相对较小的$ p $。在本文中，我们表明，对于所有$ p \ leq \ log 3/\ log 2 $和所有奇数$ n \ geq 3 $，此功能的唯一最小值。此外，我们提出了一些数值结果，表明这些最小化器的相变现象的出现。更具体地说，对于$ n \ geq 3 $奇数，存在一系列点数$ \ log 3/\ log 2 = p_1 <p_1 <p_2 <\ cdots <p_n \ leq 2 $，以便在每个子Intervals $（p_k，p_k，p_k，p_ {k+1} $上，一个唯一的（最大（某些isOmetries）最小化）在每个子INTERVALS上都存在。 ％此外，我们推测$ \ lim_ {k \ to \ infty} p_ {2k+1} = 2 $。
Given $N$ points $X=\{x_k\}_{k=1}^N$ on the unit circle in $\mathbb{R}^2$ and a number $0\leq p \leq \infty$ we investigate the minimizers of the functional $\sum_{k, \ell =1}^N |\langle x_k, x_\ell\rangle|^p$. While it is known that each of these minimizers is a spanning set for $\mathbb{R}^2$, less is known about their number as a function of $p$ and $N$ especially for relatively small $p$. In this paper we show that there is unique minimum for this functional for all $p\leq \log 3/\log 2$ and all odd $N\geq 3$. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for $N\geq 3$ odd, there exists a sequence of number of points $\log 3/\log 2=p_1< p_2< \cdots < p_N\leq 2$ so that a unique (up to some isometries) minimizer exists on each sub-intervals $(p_k, p_{k+1})$. %In addition we conjecture that $\lim_{k\to \infty}p_{2k+1}=2$.
