*Accepted Paper*

**Inserted:** 22 jan 2016

**Last Updated:** 28 aug 2016

**Journal:** Math. Models Methods Appl. Sci.

**Year:** 2016

**Abstract:**

Graphene samples are identified as minimizers of configurational energies featuring both two- and three-body atomic-interaction terms. This variational viewpoint allows for a detailed description of ground-state geometries as connected subsets of a regular hexagonal lattice. We investigate here how these geometries evolve as the number $n$ of carbon atoms in the graphene sample increases. By means of an equivalent characterization of minimality via a discrete isoperimetric inequality, we prove that ground states converge to the ideal hexagonal Wulff shape as $n\to+\infty$. Precisely, ground states deviate from such hexagonal Wulff shape by at most $K n^{3/4} + o(n^{3/4})$ atoms, where both the constant $K$ and the rate $n^{3/4}$ are sharp.

**Download:**