The dimension of the Incipient Infinite Cluster

TL;DR

This paper investigates the geometric properties of the Incipient Infinite Cluster in high-dimensional percolation, establishing that its mass dimension is almost surely 4 and its volume growth exponent is almost surely 2.

Contribution

It provides rigorous proofs that the mass dimension and volume growth exponent of the IIC are almost surely 4 and 2 respectively in high dimensions.

Findings

Mass dimension of IIC is almost surely 4.

Volume growth exponent of IIC is almost surely 2.

Results hold for high-dimensional bond percolation on z^d.

Abstract

We study the Incipient Infinite Cluster (IIC) of high-dimensional bond percolation on $\mathbb{Z}^d$. We prove that the mass dimension of IIC almost surely equals $4$ and the volume growth exponent of IIC almost surely equals $2$.