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 R² and V:R₊^*→ R is a function. For the Heitmann-Radin sticky disk potential V=VHR, we prove, using Brass' key result from [Computational Geometry, 6:195--214, 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 Z². Moreover, we numerically investigate the minimization problem for the energy per point among lattices for the Lennard-Jones potential V=VLJ:r↦ r⁻¹²-2r⁻⁶ as well as the Epstein zeta function associated to a p-norm \|·\|ₚ, i.e. when V=Vₛ:r↦ r⁻ˢ, $s>2$. Our simulations show a new and unexpected phase transition for the minimizers with respect to p.