Exponential extinction time of the contact process on finite graphs

Thomas Mountford, Jean Christophe Mourrat, Daniel Valesin, Qiang Yao

Research output: Contribution to journalArticlepeer-review


We study the extinction time τ of the contact process started with full occupancy on finite trees of bounded degree. We show that, if the infection rate is larger than the critical rate for the contact process on ℤ, then, uniformly over all trees of degree bounded by a given number, the expectation of τ grows exponentially with the number of vertices. Additionally, for any increasing sequence of trees of bounded degree, τ divided by its expectation converges in distribution to the unitary exponential distribution. These results also hold if one considers a sequence of graphs having spanning trees with uniformly bounded degree, and provide the basis for powerful coarse-graining arguments. To demonstrate this, we consider the contact process on a random graph with vertex degrees following a power law. Improving a result of Chatterjee and Durrett (2009), we show that, for any non-zero infection rate, the extinction time for the contact process on this graph grows exponentially with the number of vertices.

Original languageEnglish (US)
Pages (from-to)1974-2013
Number of pages40
JournalStochastic Processes and their Applications
Issue number7
StatePublished - Jul 2016


  • Contact process
  • Interacting particle systems
  • Metastability

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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