Supercritical Site Percolation on the Hypercube: Small Components are Small

TL;DR

This paper proves that in supercritical site percolation on a hypercube, all components except the giant are typically small, of size proportional to the dimension, resolving a long-standing conjecture.

Contribution

It establishes that all components other than the giant are of size O(d), confirming a conjecture from 1994.

Findings

All components besides the giant are of size O(d)

Resolves a conjecture from 1994

Provides a detailed analysis of supercritical percolation on hypercubes

Abstract

We consider supercritical site percolation on the $d$-dimensional hypercube $Q^d$. We show that typically all components in the percolated hypercube, besides the giant, are of size $O(d)$. This resolves a conjecture of Bollob\'as, Kohayakawa, and {\L}uczak from 1994.