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 \(0 < \lim inf_{b \to \infty} b^{\theta/(\theta-1)} p_{\rm c}({\mathbb T}_b,\theta,\infty)\leq \lim sup_{b \to \infty}b^{\theta/(\theta-1)} p_{\rm c}({\mathbb T}_b,\theta,\infty) < \infty\) .
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The work of L.R.F. was partially supported by the Brazilian CNPq through grants 307978/2004-4 and 475833/2003-1, and by FAPESP through grant 04/07276-2. The work of R.H.S. was partially supported by the American N.S.F. through grant DMS-0300672.
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Fontes, L.R., Schonmann, R.H. Threshold θ ≥ 2 contact processes on homogeneous trees. Probab. Theory Relat. Fields 141, 513–541 (2008). https://doi.org/10.1007/s00440-007-0092-z
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DOI: https://doi.org/10.1007/s00440-007-0092-z
Keywords
- Threshold Contact process
- Homogeneous trees
- Critical points
- Critical density
- Phase diagram
- Discontinuous transition
Mathematics Subject Classification (2000)
- Primary: 60K35