Climbing up a random subgraph of the hypercube

Abstract: Let $Q^d$ be the $d$-dimensional binary hypercube. We say that $P={v_1,\ldots, v_k}$ is an increasing path of length $k-1$ in $Q^d$, if for every $i\in [k-1]$ the edge $v_iv_{i+1}$ is obtained by switching some zero coordinate in $v_i$ to a one coordinate in $v_{i+1}$. Form a random subgraph $Q^d_p$ by retaining each edge in $E(Q^d)$ independently with probability $p$. We show that there is a phase transition with respect to the length of a longest increasing path around $p=\frac{e}{d}$. Let $\alpha$ be a constant and let $p=\frac{\alpha}{d}$. When $\alpha<e$, then there exists a $\delta \in [0,1)$ such that whp a longest increasing path in $Q^d_p$ is of length at most $\delta d$. On the other hand, when $\alpha>e$, whp there is a path of length $d-2$ in $Q^d_p$, and in fact, whether it is of length $d-2, d-1$, or $d$ depends on whether the all-zero and all-one vertices percolate or not.