Percolation, first-passage percolation, and covering times for Richardson's model on the n-cube

2004/04/01 by James Allen Fill, Robin Pemantle
Mathematics · #math.PR #msc:60K35 #msc:60C05

paper · pdf

published as Ann. Appl. Prob., 3, 593 - 629 (1993) · 51 pages

arxiv created 2004/04/01 · arxiv updated 2009/12/01

Abstract

Percolation with edge-passage probability p and first-passage percolation are studied for the n-cube B_n =0,1ⁿ 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->infty. This resolves a conjecture of David 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->infty.

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