Diameter and mixing time of the giant component in the percolated hypercube

Abstract

We consider bond percolation on the $d$-dimensional binary hypercube with $p=c/d$ for fixed $c>1$. We prove that the typical diameter of the giant component $L_1$ is of order $\Theta(d)$, and the typical mixing time of the lazy random walk on $L_1$ is of order $\Theta(d^2)$. This resolves long-standing open problems of Bollob\'as, Kohayakawa and {\L}uczak from 1994, and of Benjamini and Mossel from 2003. A key component in our approach is a new tight large deviation estimate on the number of vertices in $L_1$ whose proof includes several novel ingredients: a structural description of the residue outside the giant component after sprinkling, a tight quantitative estimate on the spread of the giant in the hypercube, and a stability principle which rules out the disintegration of large connected sets under thinning. This toolkit further allows us to obtain optimal bounds on the expansion in $L_1$.