documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{3/4}$$\end{document}N3/4 Law in the Cubic Lattice

• Published in: Journal of statistical physics Vol. 176; no. 6; pp. 1480 - 1499
• Main Authors: Mainini, Edoardo, Piovano, Paolo, Schmidt, Bernd, Stefanelli, Ulisse
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2019
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-019-02350-z

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_n$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{O}(n^{3/4})$$\end{document} 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.