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

Shirshendu Chatterjee and Rick Durrett

Stochastic Processes and their Applications, 2013, vol. 123, issue 2, 561-578

Abstract: 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.

Keywords: Threshold contact process; Random regular graphs; Isoperimetric inequality; First order phase transition; Binomial large deviations (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1016/j.spa.2012.10.001

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