# Fractal Geometry and Stochastics VIFractal Dimension of Discrete Sets and Percolation

Fractal Geometry and Stochastics VI: Fractal Dimension of Discrete Sets and Percolation [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Fractal Geometry and Stochastics VIFractal Dimension of Discrete Sets and Percolation

Part of the Progress in Probability Book Series (volume 76)
Editors: Freiberg, Uta; Hambly, Ben; Hinz, Michael; Winter, Steffen
23 pages

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# References (55)

Publisher
Springer International Publishing
© Springer Nature Switzerland AG 2021
ISBN
978-3-030-59648-4
Pages
101 –124
DOI
10.1007/978-3-030-59649-1_5
Publisher site
See Chapter on Publisher Site

### Abstract

[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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