Maximal fluctuations on periodic lattices: an approach via quantitative Wulff inequalities

We consider the Wulff problem arising from the study of the Heitmann-Radin energy of N atoms sitting on a periodic lattice. Combining the sharp quantitative Wulff inequality in the continuum setting with a notion of quantitative closeness between discrete and continuum energies, we provide very short proofs of fluctuation estimates of Voronoi-type sets associated with almost minimizers of the discrete problem about the continuum limit Wulff shape. In the particular case of exact energy minimizers, we recover the well-known, sharp N^3/4 scaling law for all considered planar lattices, as well as a sub-optimal scaling law for the cubic lattice in dimension d >= 3.