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May, 1993 Percolation, First-Passage Percolation and Covering Times for Richardson's Model on the $n$-Cube
James Allen Fill, Robin Pemantle
Ann. Appl. Probab. 3(2): 593-629 (May, 1993). DOI: 10.1214/aoap/1177005440

Abstract

Percolation with edge-passage probability $p$ and first-passage percolation are studied for the $n$-cube $\mathscr{B}_n = \{0, 1\}^n$ 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 $n \rightarrow \infty$. 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 $n \rightarrow \infty$.

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James Allen Fill. Robin Pemantle. "Percolation, First-Passage Percolation and Covering Times for Richardson's Model on the $n$-Cube." Ann. Appl. Probab. 3 (2) 593 - 629, May, 1993. https://doi.org/10.1214/aoap/1177005440

Information

Published: May, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0783.60102
MathSciNet: MR1221168
Digital Object Identifier: 10.1214/aoap/1177005440

Subjects:
Primary: 60K35
Secondary: 60C05

Keywords: $n$-cube , broadcasting , First-passage percolation , large deviations , Oriented percolation , percolation , Richardson's model

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.3 • No. 2 • May, 1993
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