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Register a new userExponential rate for the contact process extinction time
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We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.
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We show that the contact process on the rank-one inhomogeneous random graphs and Erdos-R{e}nyi graphs with mean degree large enough survives a time exponential in the size of these graphs for any positive infection rate. In addition, a metastable result for the extinction time is also proved.
The logistic birth and death process is perhaps the simplest stochastic population model that has both density-dependent reproduction, and a phase transition, and a lot can be learned about the process by studying its extinction time, $tau_n$, as a function of system size $n$. A number of existing results describe the scaling of $tau_n$ as $ntoinfty$, for various choices of reproductive rate $r_n$ and initial population $X_n(0)$ as a function of $n$. We collect and complete this picture, obtaining a complete classification of all sequences $(r_n)$ and $(X_n(0))$ for which there exist rescaling parameters $(s_n)$ and $(t_n)$ such that $(tau_n-t_n)/s_n$ converges in distribution as $ntoinfty$, and identifying the limits in each case.
In this note we give a new method for getting a series of approximations for the extinction probability of the one-dimensional contact process by using the Grobner basis.
We study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $lambda_1$ for weak survival, and the survival probability $p(lambda)$ is continuous with respect to the infection rate $lambda$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $lambda_1<lambda_2$, which confirms a conjecture of Staceys cite{Stacey}. We also prove that if the contact process survives strongly at $lambda$ then it survives strongly at a $lambda<lambda$, which implies that the process does not survive strongly at the critical value $lambda_2$ for strong survival.
We construct graphs (trees of bounded degree) on which the contact process has critical rate (which will be the same for both global and local survival) equal to any prescribed value between zero and $lambda_c(mathbb{Z})$, the critical rate of the one-dimensional contact process. We exhibit both graphs in which the process at this target critical value survives (locally) and graphs where it dies out (globally).
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