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

created by leonardi on 21 Jan 2019

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Inserted: 21 jan 2019
Last Updated: 21 jan 2019

Year: 2019

Abstract:

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 $d\ge 3$.