## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 3, Number 2 (1993), 593-629.

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

James Allen Fill and Robin Pemantle

#### 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 $n \rightarrow \infty$.

#### Article information

**Source**

Ann. Appl. Probab., Volume 3, Number 2 (1993), 593-629.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005440

**Digital Object Identifier**

doi:10.1214/aoap/1177005440

**Mathematical Reviews number (MathSciNet)**

MR1221168

**Zentralblatt MATH identifier**

0783.60102

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60C05: Combinatorial probability

**Keywords**

Richardson's model $n$-cube percolation oriented percolation first-passage percolation large deviations broadcasting

#### Citation

Fill, James Allen; Pemantle, Robin. Percolation, First-Passage Percolation and Covering Times for Richardson's Model on the $n$-Cube. Ann. Appl. Probab. 3 (1993), no. 2, 593--629. doi:10.1214/aoap/1177005440. https://projecteuclid.org/euclid.aoap/1177005440