### Abstract

We consider the contact process with infection rate λ on T_{n}^{d}, the d-ary tree of height n. We study the extinction time τ_{Tnd}, that is, the random time it takes for the infection to disappear when the process is started from full occupancy. We prove two conjectures of Stacey regarding τ_{Tnd}. Let λ_{2} denote the upper critical value for the contact process on the infinite d-ary tree. First, if λ < λ_{2}, then τT^{d} _{n} divided by the height of the tree converges in probability, as n → ∞, to a positive constant. Second, if λ > λ_{2}, then log E[τ_{Tnd}] divided by the volume of the tree converges in probability to a positive constant, and τ_{Tnd}/E[τ_{Tnd}] converges in distribution to the exponential distribution of mean 1.

Original language | English (US) |
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Pages (from-to) | 385-408 |

Number of pages | 24 |

Journal | Alea |

Volume | 11 |

Issue number | 1 |

State | Published - Jan 1 2014 |

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### Keywords

- Contact process
- Interacting particle systems
- Metastability

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Alea*,

*11*(1), 385-408.