Uniform multicommodity flow in the hypercube with random edge capacities

We give two results for multicommodity flows in the d-dimensional hypercube Qd with independent random edge capacities distributed like C where Pr[C>0]>1/2. Firstly, with high probability as d rightarrow ∞, 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 21-d E[C] between all vertex pairs. Both results are best possible.