UNORIENTED FIRST-PASSAGE PERCOLATION ON THE n-CUBE BY ANDERS MARTINSSON

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 Tn of a path from (0,0, . . . ,0) to (1,1, . . . ,1) converges in probability to ln(1 +√2)≈ 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+√2)− o(1)≤ Tn ≤ 1+ o(1), and has been conjectured to converge in probability by Bollobás 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 Tn ≥ ln(1 + √ 2)− o(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.