PHASE TRANSITIONS FOR THE MINIMIZERS OF THE p-FRAME POTENTIALS IN R^2*

Given N points (Formula Presented) on the unit circle in R² and a number 0 ≤ p ≤ ∞, we investigate the minimizers of the functional (Formula Presented). While it is known that each of these minimizers is a spanning set for R², 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 ≤ log 3/log 2 and all odd N ≥ 3. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for N ≥ 3 odd, there exists a sequence of points log 3/log 2 = p1 < p2 < ••• < p_N ≤ 2 so that a unique (up to some isometries) minimizer exists on each of the subintervals (p_k, pk₊₁).