Fractal dimension of discrete sets and percolation

Abstract

There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of ${\mathbb{R}}^d$. In this expository text, we discuss their analogues for infinite subsets of ${\mathbb{Z}}^d$ and, more generally, for infinite graphs. We then apply these notions to critical percolation clusters, where the various dimensions have different values.

Expository text, intended for Proceedings of Fractal Geometry and Stochastics VI. Version v2: improved Lemma 1.1