We consider the contact process with infection rate λ on Tnd, the d-ary tree of height n. We study the extinction time τTnd, 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 τTnd. Let λ2 denote the upper critical value for the contact process on the infinite d-ary tree. First, if λ < λ2, then τTd n divided by the height of the tree converges in probability, as n → ∞, to a positive constant. Second, if λ > λ2, then log E[τTnd] divided by the volume of the tree converges in probability to a positive constant, and τTnd/E[τTnd] converges in distribution to the exponential distribution of mean 1.
|Original language||English (US)|
|Number of pages||24|
|State||Published - Jan 1 2014|
- Contact process
- Interacting particle systems
ASJC Scopus subject areas
- Statistics and Probability