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

  • Marco CicaleseEmail author
  • Gian Paolo Leonardi
Article
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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 \(d\ge 3\).

Notes

Acknowledgements

The work of MC was partially supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”. GP has been partially supported by the INdAM-GNAMPA Project 2019 "Problemi isoperimetrici in spazi euclidei e non". Part of this work was carried out while GP was visiting the department of mathematics of TUM, whose hospitality is gratefully acknowledged.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Zentrum Mathematik - M7Technische Universität MünchenGarchingGermany
  2. 2.Department of MathematicsUniversity of TrentoPovoItaly

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