Percolation, First-Passage Percolation and Covering Times for Richardson's Model on the n-Cube

Percolation with edge-passage probability p and first-passage percolation are studied for the n-cube Bn={0,1}n with nearest neighbor edges. For oriented and unoriented percolation, p=e/n and p=1/n are the respective critical probabilities. For oriented first-passage percolation with i.i.d. edge-passage times having a density of 1 near the origin, the percolation time (time to reach the opposite corner of the cube) converges in probability to 1 as n... This resolves a conjecture of Aldous. When the edge-passage distribution is standard exponential, the (smaller) percolation time for unoriented edges is at least 0.88. These results are applied to Richardson's model on the (unoriented) n-cube. Richardson's model, otherwise known as the contact process with no recoveries, models the spread of infection as a Poisson process on each edge connecting an infected node to an uninfected one. It is shown that the time to cover the entire n-cube is bounded between 1.41 and 14.05 in probability as n...