Maximal Fluctuations Around the Wulff Shape for Edge-Isoperimetric Sets in Zd: A Sharp Scaling Law

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice Z d from the limiting Wulff shape in arbitrary dimensions. As the number n of elements diverges, we prove that the symmetric difference to the corresponding Wulff set consists of at most O ( n ( d - 1 + 2 1 - d )...

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Bibliographic Details
Published inCommunications in mathematical physics Vol. 380; no. 2; pp. 947 - 971
Main Authors Mainini, Edoardo, Schmidt, Bernd
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 2020
Springer Nature B.V
Subjects
Classical and Quantum Gravitation
Complex Systems
Constraining
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Scaling laws
Theoretical
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Summary:We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice Z d from the limiting Wulff shape in arbitrary dimensions. As the number n of elements diverges, we prove that the symmetric difference to the corresponding Wulff set consists of at most O ( n ( d - 1 + 2 1 - d ) / d ) lattice points and that the exponent ( d - 1 + 2 1 - d ) / d is optimal. This extends the previously found ‘ n 3 / 4 laws’ for d = 2 , 3 to general dimensions. As a consequence we obtain optimal estimates on the rate of convergence to the limiting Wulff shape as n diverges.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-020-03879-x