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Exponential extinction time of the contact process on finite graphs

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

Abstract

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.

Suggested Citation

  • Mountford, Thomas & Mourrat, Jean-Christophe & Valesin, Daniel & Yao, Qiang, 2016. "Exponential extinction time of the contact process on finite graphs," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 1974-2013.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:7:p:1974-2013
    DOI: 10.1016/j.spa.2016.01.001
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    References listed on IDEAS

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    1. Durrett, Rick & Jung, Paul, 2007. "Two phase transitions for the contact process on small worlds," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1910-1927, December.
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    Cited by:

    1. Stein Andreas Bethuelsen & Gabriel Baptista Silva & Daniel Valesin, 2022. "Graph Constructions for the Contact Process with a Prescribed Critical Rate," Journal of Theoretical Probability, Springer, vol. 35(2), pages 863-893, June.
    2. Van Hao Can, 2019. "Exponential Extinction Time of the Contact Process on Rank-One Inhomogeneous Random Graphs," Journal of Theoretical Probability, Springer, vol. 32(1), pages 106-130, March.

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