Critical phase transitions in minimum-energy configurations for the exponential kernel family $e^{-|x-y|^q}$ on the unit interval

Abstract

We study the optimal placement of $k$ ordered points on the unit interval for the bounded pair potential \[ K_q(d)=e^{-d^q}, \qquad q>0. \] The family interpolates between strongly cusp-like kernels for $0<q<1$, the threshold kernel $e^{-d}$, and the flatter Gaussian-type regime $q>1$. Our emphasis is on the transition from collision-free minimizers to endpoint-collapsed minimizers. We reformulate the problem in gap variables, record convexity, symmetry, and the Karush-Kuhn-Tucker conditions, and give a short proof that collisions are impossible for $0<q<1$. At the threshold $q=1$ we recover the endpoint-clustering law for $e^{-d}$, while for $q>1$ we identify critical exponents $q_k$ beyond which interior points are no longer optimal. For odd $k$ we derive the exact universal value \[ q_{2m+1} = \frac{\log(1/(-\log((1+e^{-1})/2)))}{\log 2} \approx 1.396363475, \] and for even $k=4,6,\dots,20$ we compute the numerical transition values \[ \begin{aligned} &q_4\approx 1.062682507,\quad q_6\approx 1.155601329,\quad q_8\approx 1.206132611,\quad q_{10}\approx 1.238523533,\\ &q_{12}\approx 1.261308114,\quad q_{14}\approx 1.278305167,\quad q_{16}\approx 1.291510874,\quad q_{18}\approx 1.302082885,\\ &q_{20}\approx 1.310744185. \end{aligned} \] We also include comparison tables and diagrams for the kernels $e^{-\sqrt d}$, $e^{-d}$, and $e^{-d^2}$, briefly relate the bounded family to the singular Riesz kernel $d^{-s}$, and identify the $q\to 0^+$ limit with the Fekete/Chebyshev--Lobatto configuration on $[0,1]$.