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

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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 \(d\ge 3\).

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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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Authors and Affiliations

  1. Zentrum Mathematik - M7, Technische Universität München, Garching, Germany

    Marco Cicalese

  2. Department of Mathematics, University of Trento, 38050, Povo, TN, Italy

    Gian Paolo Leonardi

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  1. Marco Cicalese
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  2. Gian Paolo Leonardi
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Correspondence to Marco Cicalese.

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Communicated by H. Duminil-Copin.

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Cicalese, M., Leonardi, G.P. Maximal Fluctuations on Periodic Lattices: An Approach via Quantitative Wulff Inequalities. Commun. Math. Phys. 375, 1931–1944 (2020). https://doi.org/10.1007/s00220-019-03612-3

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