Fractal dimension of discrete sets and percolation

TL;DR

This paper explores different notions of fractal dimension for infinite sets in lattices and graphs, and applies these concepts to analyze the structure of critical percolation clusters.

Contribution

It introduces and discusses fractal dimensions for discrete sets and graphs, and applies these ideas to understand critical percolation clusters in detail.

Findings

Different fractal dimensions have distinct values for percolation clusters

The notions extend classical fractal geometry to discrete and graph settings

Applications to critical phenomena in percolation theory

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.