The Mihail-Vazirani conjecture and strong edge-expansion in random $0/1$ polytopes

We study the edge-expansion of the graph of a random $0/1$ polytope $P^d_p$, defined as the convex hull of a random subset of the points in $\{0,1\}^d$ where every point is retained independently and with probability $p$. This problem was introduced more than twenty years ago in a work of Gillmann and Kaibel, and has been extensively studied ever since. We prove that, for every fixed $\varepsilon>0$ and every $p\in(0,1-\varepsilon]$, with high probability the graph of $P^d_p$ has edge-expansion $\Theta(d)$. This improves the previously best known bound due to Ferber, Krivelevich, Sales and Samotij, and verifies, in a strong form, the celebrated Mihail-Vazirani conjecture for random $0/1$ polytopes. Although the expansion factor $\Theta(d)$ is typically best possible for $p\ge 1/2+\varepsilon$, we also show that the behaviour changes drastically at $p=1/2$. Namely, for every fixed $\varepsilon>0$ and every integer $k\ge 2$, if $p\le 1/2-\varepsilon$, then with high probability the graph of $P^d_p$ has edge-expansion $\Omega(d^k)$. Thus, random $0/1$ polytopes exhibit an interesting phase transition at $p=1/2$.