Calculus of Variations and Geometric Measure Theory

M. Cicalese - G. P. Leonardi

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

created by leonardi on 21 Jan 2019
modified by cicalese on 02 Apr 2021


Published Paper

Inserted: 21 jan 2019
Last Updated: 2 apr 2021

Journal: Commun. Math. Phys.
Volume: 375
Pages: 1931–1944
Year: 2019


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 $d\ge 3$.