Threshold $theta geq 2$ contact processes on homogeneous trees
  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 $0 < liminf_{b to infty} b^{theta(theta-1)} p_c(T_b,theta,infty) leq limsup_{b to infty} b^{theta/(theta-1)} p_c(T_b,theta,infty) < infty$. 
Luiz Renato Fontes and Roberto H. Schonmann