Threshold θ ≥ 2 contact processes on homogeneous trees

We study the threshold θ ≥ 2 contact process on a homogeneous tree $$\mathbb T_b$$ of degree κ = b + 1, with infection parameter λ ≥ 0 and started from a product measure with density p. The corresponding mean-field model displays a discontinuous transition at a critical point $$\lambda_{\rm c}^{\rm MF}(\kappa,\theta)$$ and for $$\lambda \geq\lambda_{\rm c}^{\rm MF}(\kappa,\theta)$$ it survives iff $$p \geq p_{\rm c}^{\rm MF}(\kappa,\theta,\lambda)$$ , where this critical density satisfies $$0 < p_{\rm c}^{\rm MF}(\kappa,\theta,\lambda) < 1$$ , $$ \lim_{\lambda \to \infty} p_{\rm c}^{\rm MF}(\kappa,\theta,\lambda) = 0$$ . For large b, we show that the process on $$\mathbb T_b$$ has a qualitatively similar behavior when λ is small, including the behavior at and close to the critical point $$\lambda_{\rm c}(\mathbb T_b,\theta)$$ . In contrast, for large λ the behavior of the process on $$\mathbb T_b$$ is qualitatively distinct from that of the mean-field model in that the critical density has $$p_{\rm c}({\mathbb T}_b,\theta,\infty)\,:= \lim_{\lambda \to \infty} p_{\rm c}(\mathbb T_b,\theta,\lambda) > 0$$ . We also show that $$\lim_{b \to \infty} b \lambda_{\rm c}(\mathbb T_b,\theta) = \Phi_{\theta}$$ , where 1 < Φ2 < Φ3 < ..., $$\lim_{\theta \to \infty} \Phi_{\theta} = \infty$$ , and $$0 < \lim inf_{b \to \infty} b^{\theta/(\theta-1)} p_{\rm c}({\mathbb T}_b,\theta,\infty)\leq \lim sup_{b \to \infty}b^{\theta/(\theta-1)} p_{\rm c}({\mathbb T}_b,\theta,\infty) < \infty$$ .