UNORIENTED FIRST-PASSAGE PERCOLATION ON THE n-CUBE

The n-dimensional binary hypercube is the graph whose vertices are the binary n-tuples {0,1)(n) and where two vertices are connected by an edge if they differ at exactly one coordinate. We prove that if the edges are assigned independent mean 1 exponential costs, the minimum length T-n of a path from (0,0,..., 0) to (1,1,, 1) converges in probability to ln(1 + root 2) approximate to 0.881. It has previously been shown by Fill and Pemantle [Ann. Appl. Probab. 3 (1993) 593-629] that this so-called first-passage time asymptotically almost surely satisfies ln(1 + root 2) - 0(1) <= T-n <= 1+ 0(1), and has been conjectured to converge in probability by Bollobas and Kohayakawa [In Combinatorics, Geometry and Probability (Cambridge, 1993) (1997) 129-137 Cambridge]. A key idea of our proof is to consider a lower bound on Richardson's model, closely related to the branching process used in the article by Fill and Pemantle to obtain the bound T-n >= ln(1 + root 2) - 0(1). We derive an explicit lower bound on the probability that a vertex is infected at a given time. This result is formulated for a general graph and may be applicable in a more general setting.