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 $0\le p\le \mathrm{\infty }$ we investigate the minimizers of the functional ${\sum }_{k,\mathrm{\ell }=1}^N\mathrm{\mid }⟨x_k,x_{\mathrm{\ell }}⟩{\mathrm{\mid }}^p$. 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 $p\le \log 3\mathrm{/}\log 2$ and all odd $N\ge 3$. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for $N\ge 3$ odd, there exists a sequence of number of points $\log 3\mathrm{/}\log 2=p_1<p_2<\cdots <p_N\le 2$ so that a unique (up to some isometries) minimizer exists on each sub-intervals $(p_k,p_{k+1})$. %In addition we conjecture that ${\lim }_{k\to \mathrm{\infty }}{p}_{2k+1}=2$.