Low-dimensional faces of random 0/1-polytopes
Let P be a random $d$-dimensional 0/1-polytope with $n(d)$ vertices, and denote by $ϕ_k(P)$ the \emph{$k$-face density} of $P$, i.e., the quotient of the number of $k$-dimensional faces of $P$ and $\binom{n(d)}{k+1}$. For each $k\ge 2$, we establish the existence of a sharp threshold for the $k$-face density and determine the values of the threshold numbers $τ_k$ such that, for all $ε>0$, $$ E(ϕ_k(P)) = \begin{cases} 1-o(1) & \text{if $n(d)\le 2^{(τ_k-ε)d}$ for all $d$} o(1) & \text{if $n(d)\ge 2^{(τ_k+ε)d}$ for all $d$} \end{cases} $$ holds for the expected value of $ϕ_k(P)$. The threshold for $k=1$ has recently been determined in \texttt{math.CO/0306246}. In particular, these results indicate that the high face densities often encountered in polyhedral combinatorics (e.g., for the cut-polytopes of complete graphs) should be considered more as a phenomenon of the general geometry of 0/1-polytopes than as a feature of the special combinatorics of the underlying problems.