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A first order phase transition in the threshold θ≥2 contact process on random r-regular graphs and r-trees


  • Chatterjee, Shirshendu
  • Durrett, Rick


We consider the discrete time threshold-θ contact process on a random r-regular graph. We show that if θ≥2, r≥θ+2, ϵ1 is small and p≥p1(ϵ1), then starting from all vertices occupied the fraction of occupied vertices is ≥1−2ϵ1 up to time exp(γ1(r)n) with high probability. We also show that for p2<1 there is an ϵ2(p2)>0 so that if p≤p2 and the initial density is ≤ϵ2(p2), then the process dies out in time O(logn). These results imply that the process on the r-tree has a first-order phase transition.

Suggested Citation

  • Chatterjee, Shirshendu & Durrett, Rick, 2013. "A first order phase transition in the threshold θ≥2 contact process on random r-regular graphs and r-trees," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 561-578.
  • Handle: RePEc:eee:spapps:v:123:y:2013:i:2:p:561-578 DOI: 10.1016/

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    References listed on IDEAS

    1. Sztonyk, Pawel, 2011. "Transition density estimates for jump Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1245-1265, June.
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