On crystallization in the plane for pair potentials with an arbitrary norm

We investigate two-dimensional crystallization phenomena, i.e., minimality of a lattice’s patch for interaction energies, with pair potentials of type ( x , y ) ↦ V ( ∥ x - y ∥ ) where ∥ · ∥ is an arbitrary norm on ℝ 2 and V : ℝ + * → ℝ is a function. For the Heitmann–Radin sticky disk potential V = V H R , we prove, using Brass’ key result from Computational Geometry 1996, that crystallization occurs for any fixed norm, with a classification of minimizers and minimal energies according to the kissing number associated to ∥ · ∥ . The minimizer is proved to be, up to affine transform, a patch of the triangular or the square lattice, which shows how to easily get anisotropy in a crystallization phenomenon. We apply this result to the p -norms ∥ · ∥ p , p ≥ 1 , which allows us to construct an explicit family of norms for which crystallization holds on any given lattice. We also solve part of a crystallization problem studied in [ Arch. Ration. Mech. Anal. , 240:987–1053] where points are constrained to be on ℤ 2 . Moreover, we numerically investigate the minimization problem for the energy per point among lattices for the Lennard–Jones potential V = V L J : r ↦ r - 12 - 2 r - 6 as well as the Epstein zeta function associated to a p -norm ∥ · ∥ p , i.e., when V = V s : r ↦ r - s , s > 2 . Our simulations show a new and unexpected phase transition for the minimizers with respect to p .