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Threshold θ ≥ 2 contact processes on homogeneous trees.
- Source :
-
Probability Theory & Related Fields . Jul2008, Vol. 141 Issue 3/4, p513-541. 29p. - Publication Year :
- 2008
-
Abstract
- 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 $$0 < \lim inf_{b \to \infty} b^{\theta/(\theta-1)} p_{\rm c}({\mathbb T}_b,\theta,\infty)\leq \lim sup_{b \to \infty}b^{\theta/(\theta-1)} p_{\rm c}({\mathbb T}_b,\theta,\infty) < \infty$$ . [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01788051
- Volume :
- 141
- Issue :
- 3/4
- Database :
- Academic Search Index
- Journal :
- Probability Theory & Related Fields
- Publication Type :
- Academic Journal
- Accession number :
- 31587115
- Full Text :
- https://doi.org/10.1007/s00440-007-0092-z