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

arXiv2212.04444MaRDI QIDQ6420001

Radel Ben-Av, Xuemei Chen, Assaf Goldberger, Author name not available (Why is that?), Kasso A. Okoudjou

Publication date: 8 December 2022

Abstract: Given $N$ points $X={x_k}_{k=1}^N$ on the unit circle in $mathbb{R}^2$ and a number $0leqpleqinfty$ we investigate the minimizers of the functional $su{m}_{k,ell=1}^N|langle{x}_k,x_{e}llangle{|}^p$. 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 $pleqlog3/log2$ and all odd $Ngeq3$. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for $Ngeq3$ odd, there exists a sequence of number of points $log3/log2=p_1<p_2<cdots<p_{N}leq2$ so that a unique (up to some isometries) minimizer exists on each sub-intervals $(p_k,p_{k+1})$. %In addition we conjecture that $li{m}_{koinfty}{p}_{2k+1}=2$.

Mathematics Subject Classification ID

Inequalities and extremum problems involving convexity in convex geometry (52A40) General harmonic expansions, frames (42C15) Packing and covering in (n) dimensions (aspects of discrete geometry) (52C17)

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