Uniform multicommodity flow in the hypercube with random edge capacities

We give two results for multicommodity flows in the $d$-dimensional hypercube
${Q}^d$ with independent random edge capacities distributed like $C$ where
$\Pr[C>0]>1/2$. Firstly, with high probability as $d \rightarrow \infty$, the
network can support simultaneous multicommodity flows of volume close to $E[C]$
between all antipodal vertex pairs. Secondly, with high probability, the
network can support simultaneous multicommodity flows of volume close to
$2^{1-d} E[C]$ between all vertex pairs. Both results are best possible.