Expansion, long cycles, and complete minors in supercritical random subgraphs of the hypercube

Analogous to the case of the binomial random graph $G(d+1,p)$, it is known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, which we denote by $Q^d_p$, undergoes a phase transition around the critical value of $p=\frac{1}{d}$. More precisely, standard arguments show that significantly below this value of $p$, with probability tending to one as $d \to \infty$ (whp for short) all components of this graph have order $O(d)$, whereas Ajtai, Koml\'{o}s and Szemer\'{e}di showed that significantly above this value, in the \emph{supercritical regime}, whp there is a unique `giant' component of order $\Theta\left(2^d\right)$. In $G(d+1,p)$ much more is known about the complex structure of the random graph which emerges in this supercritical regime. For example, it is known that in this regime whp $G(d+1,p)$ contains paths and cycles of length $\Omega(d)$, as well as complete minors of order $\Omega\left(\sqrt{d}\right)$. In this paper we obtain analogous results in $Q^d_p$. In particular, we show that for supercritical $p$, i.e., when $p=\frac{1+\epsilon}{d}$ for a positive constant $\epsilon$, whp $Q^d_p$ contains a cycle of length $\Omega\left(\frac{2^d}{d^3(\log d)^3} \right)$ and a complete minor of order $\Omega\left(\frac{2^{\frac{d}{2}}}{d^3(\log d)^3 }\right)$. In order to prove these results, we show that whp the largest component of $Q^d_p$ has good edge-expansion properties, a result of independent interest. We also consider the genus of $Q^d_p$ and show that, in this regime of $p$, whp the genus is $\Omega\left(2^d\right)$.