N-3/4 Law in the Cubic Lattice

Edoardo Mainini; Paolo Piovano; Bernd Schmidt; Ulisse Stefanelli

doi:10.1007/s10955-019-02350-z

N-3/4 Law in the Cubic Lattice

Edoardo Mainini, Paolo Piovano, Bernd Schmidt, Ulisse Stefanelli

Department of Mathematics

Publications: Contribution to journal › Article › Peer Reviewed

Abstract

We investigate the Edge-Isoperimetric Problem (EIP) for sets with n elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers M _n of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most O (n ^{3 / 4}) elements. The exponent 3 / 4 is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions.

Original language	English
Pages (from-to)	1480 - 1499
Number of pages	20
Journal	Journal of Statistical Physics
Volume	176
Issue number	6
DOIs	https://doi.org/10.1007/s10955-019-02350-z
Publication status	Published - Sep 2019

Austrian Fields of Science 2012

101002 Analysis

Keywords

CRYSTALLIZATION
Cubic lattice
DEVIATION
Edge perimeter
Fluctuations
GROUND-STATE
ISING-MODEL
N-3/4 law
Wulff shape
N law

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http://phaidra.univie.ac.at/o:1078975Licence: CC BY 4.0

Cite this

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title = "N-3/4 Law in the Cubic Lattice",

abstract = "We investigate the Edge-Isoperimetric Problem (EIP) for sets with n elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers M n of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most O (n 3 / 4) elements. The exponent 3 / 4 is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions. ",

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AU - Piovano, Paolo

AU - Schmidt, Bernd

AU - Stefanelli, Ulisse

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AB - We investigate the Edge-Isoperimetric Problem (EIP) for sets with n elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers M n of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most O (n 3 / 4) elements. The exponent 3 / 4 is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions.

KW - CRYSTALLIZATION

KW - Cubic lattice

KW - DEVIATION

KW - Edge perimeter

KW - Fluctuations

KW - GROUND-STATE

KW - ISING-MODEL

KW - N-3/4 law

KW - Wulff shape

KW - N law

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EP - 1499

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

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ER -

N-3/4 Law in the Cubic Lattice

Abstract

Austrian Fields of Science 2012

Keywords

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