Sharp $$N^{3/4}$$ Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice

We investigate the edge-isoperimetric problem (EIP) for sets of n points in the triangular lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. By introducing a suitable notion of perimeter and area, EIP minimizers are characterized as extremizers of an isoperimetric inequality: they attain maximal area and minimal perimeter among connected configurations. The maximal area and minimal perimeter are explicitly quantified in terms of n. In view of this isoperimetric characterizations, EIP minimizers (M_n) are seen to be given by hexagonal configurations with some extra points at their boundary. By a careful computation of the cardinality of these extra points, minimizers (M_n) are estimated to deviate from such hexagonal configurations by at most (K_t, n^{3/4}+mathrm{o}(n^{3/4})) points. The constant (K_t) is explicitly determined and shown to be sharp.