Phase transitions for frame potentials]{Phase transitions for the minimizers of the $p^{th}$ frame potentials in $\mathbb{R}^2$

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 $\lim_{k\to \infty}p_{2k+1}=2$.