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[There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {R}^d$$ \end{document}. In this expository text, we discuss their analogues for infinite subsets of ℤd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb {Z}^d}$$ \end{document} and, more generally, for infinite graphs. We then apply these notions to critical percolation clusters, where the various dimensions have different values.]
Published: Mar 24, 2021
Keywords: Discrete fractal; Fractal dimension; Mass dimension; Spectral dimension; Discrete Hausdorff dimension; Percolation; Incipient infinite cluster; 28A80; 60K35; 82B43
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