A discontinuous phase transition in the threshold-θ≥ 2 contact process on random graphs

2019/07/11 by Nam, Danny
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1907.05005

Abstract

We study the discrete-time threshold-θ≥ 2 contact process on random graphs of general degrees. For random graphs with a given degree distribution μ, we show that if μ is lower bounded by θ+2 and has finite kth moments for all k>0, then the discrete-time threshold-θ contact process on the random graph exhibits a discontinuous phase transition in the emergence of metastability, thus answering a question of Chatterjee and Durrett \citecd13. To be specific, we establish that (i) for any large enough infection probability p>p₁, the process started from the all-infected state w.h.p. survives for e^Θ(n)-time, maintaining a large density of infection; (ii) for any p<1, if the initial density is smaller than ε(p)>0, then it dies out in O(log n)-time w.h.p.. We also explain some extensions to more general random graphs, including the Erdős-Rényi graphs. Moreover, we prove that the threshold-θ contact process on a random (θ+1)-regular graph dies out in time n^O(1) w.h.p..

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