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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 $$N^{3/4}$$ N 3 / 4 scaling law for all considered planar lattices, as well as a sub-optimal scaling law for the cubic lattice in dimension $$d\ge 3$$ d ≥ 3 .

Maximal Fluctuations on Periodic Lattices: An Approach via Quantitative Wulff Inequalities