齐次树上的门限$theta geq 2$接触过程
Threshold $theta geq 2$ contact processes on homogeneous trees
摘要：
我们研究了$theta geq 2$接触过程的均匀树$T_b$度$kappa = b + 1$，感染参数$lambda geq 0$和开始从一个产品的测量密度$p$。相应的平均场模型在临界点$lambda_c^{MF}(kappa,theta)$处显示不连续的转变，并且对于$lambda geq lambda_c^{MF}(kappa,theta)$，它存活当且仅当$p geq p_c^{MF}(kappa,theta,lambda)$，其中该临界密度满足$0 < p_c^{MF}(kappa,theta,lambda) < 1$，$lim_{lambda to infty} p_c^{MF}(kappa,theta,lambda) = 0$。对于大的$b$，我们表明，$T_b$的过程具有定性相似的行为时，$lambda$是小的，包括在和接近的临界点$lambda_c(T_b,theta)$的行为。相比之下，对于大的$p_c(T_b,theta,infty) := lim_{lambda to infty} p_c(T_b,theta,lambda) > 0$，$T_b$上的过程的行为在性质上不同于平均场模型，因为临界密度具有$p_c(T_b,theta,infty) := lim_{lambda to infty} p_c(T_b,theta,lambda) > 0$。我们还表明，$lim_{b to infty} b lambda_c(T_b,theta) = Phi_{theta}$，其中$1 < Phi_2 < Phi_3 < ...$，$lim_{theta to infty} Phi_{theta} = infty$，和$0 < liminf_{b to infty} b^{theta(theta-1)} p_c(T_b,theta,infty) leq limsup_{b to infty} b^{theta/(theta-1)} p_c(T_b,theta,infty) < infty$。
英文摘要：
We study the threshold $theta geq 2$ contact process on a homogeneous tree $T_b$ of degree $kappa = b + 1$, with infection parameter $lambda geq 0$ and started from a product measure with density $p$. The corresponding mean-field model displays a discontinuous transition at a critical point $lambda_c^{MF}(kappa,theta)$ and for $lambda geq lambda_c^{MF}(kappa,theta)$ it survives iff $p geq p_c^{MF}(kappa,theta,lambda)$, where this critical density satisfies $0 < p_c^{MF}(kappa,theta,lambda) < 1$, $lim_{lambda to infty} p_c^{MF}(kappa,theta,lambda) = 0$. For large $b$, we show that the process on $T_b$ has a qualitatively similar behavior when $lambda$ is small, including the behavior at and close to the critical point $lambda_c(T_b,theta)$. In contrast, for large $lambda$ the behavior of the process on $T_b$ is qualitatively distinct from that of the mean-field model in that the critical density has $p_c(T_b,theta,infty) := lim_{lambda to infty} p_c(T_b,theta,lambda) > 0$. We also show that $lim_{b to infty} b lambda_c(T_b,theta) = Phi_{theta}$, where $1 < Phi_2 < Phi_3 < ...$, $lim_{theta to infty} Phi_{theta} = infty$, and $0 < liminf_{b to infty} b^{theta(theta-1)} p_c(T_b,theta,infty) leq limsup_{b to infty} b^{theta/(theta-1)} p_c(T_b,theta,infty) < infty$.
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