Inhomogeneous contact processes on trees. (English) Zbl 0924.60083
Summary: We consider an inhomogeneous contact process on a tree \(\mathbb{T}_k\) of degree \(k\), where the infection rate at any site is \(\lambda\), the death rate at any site in \(S\subset \mathbb{T}_k\) is \(\delta\) (with \(0< \delta\leq 1\)) and that at any site in \(\mathbb{T}_k- S\) is 1. Denote by \(\lambda_c (\mathbb{T}_k)\) the critical value for the homogeneous model (i.e., \(\delta=1\)) on \(\mathbb{T}_k\) and by \(\theta (\delta, \lambda)\) the survival probability of the inhomogeneous model on \(\mathbb{T}_k\). We prove that when \(k>4\), if \(S= \mathbb{T}_\sigma\), a subtree embedded in \(\mathbb{T}_k\), with \(1\leq \sigma\leq \sqrt{k}\), then there exists \(\delta_c^\sigma\) strictly between \(\lambda_c (\mathbb{T}_k)/ \lambda_c(\mathbb{T}_\sigma)\) and 1 such that \(\theta (\delta, \lambda_c (\mathbb{T}_k)) =0\) when \(\delta> \delta_c^\sigma\) and \(\theta (\delta, \lambda_c (\mathbb{T}_k))>0\) when \(\delta< \delta_c^\sigma\); if \(S= \{o\}\), the origin of \(\mathbb{T}_k\), then \(\theta (\delta, \lambda_c (\mathbb{T}_k))=0\) for any \(\delta\in (0,1)\).
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
References:
| [1] | M. Aizenman and G. R. Grimmett, Strict monotonicity for critical points in percolation and ferromagnetic models,J. Stat. Phys. 63:817–835 (1991). · doi:10.1007/BF01029985 |
| [2] | M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models,J. Stat. Phys. 36:107–143 (1984). · Zbl 0586.60096 · doi:10.1007/BF01015729 |
| [3] | D. J. Barsky and C. C. Wu, Critical exponents for the contact process under the triangle condition, Submitted toJ. Stat. Phys. (1996). · Zbl 0919.60087 |
| [4] | C. E. Bezuidenhout and G. R. Grimmett, The critical contact process dies out,Ann. Probab. 18:1462–1482 (1990). · Zbl 0718.60109 · doi:10.1214/aop/1176990627 |
| [5] | C. E. Bezuidenhout and G. R. Grimmett, Exponential decay for subcritical contact and percolation processes,Ann. Probab. 19:984–1009 (1991). · Zbl 0743.60107 · doi:10.1214/aop/1176990332 |
| [6] | G. R. Grimmett and C. M. Newman, Percolation in + 1 dimensions, InDisorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh, eds.), pp. 167–190 (Clarendon Press, Oxford, 1990). |
| [7] | T. M. Liggett, Multiple transition points for the contact process on binary trees,Ann. Probab. 24:1675–1710 (1996). · Zbl 0871.60087 · doi:10.1214/aop/1041903202 |
| [8] | N. Madras, R. Schinazi, and R. H. Schonmann, On the critical behavior of the contact process in deterministic inhomogeneous environments,Ann. Probab. 22:1140–1159 (1994). · Zbl 0821.60091 · doi:10.1214/aop/1176988598 |
| [9] | G. Morrow, R. Schinazi, and Y. Zhang, The critical contact process on a homogeneous tree,J. Appl. Probab. 31:250–255 (1994). · Zbl 0798.60091 · doi:10.2307/3215251 |
| [10] | C. M. Newman and C. C. Wu, Percolation and contact processes with low-dimensional inhomogeneity, to appear inAnn. Probab. (1997). · Zbl 0901.60074 |
| [11] | R. Pemantle, The contact process on trees,Ann. Probab. 20:2089–2116 (1992). · Zbl 0762.60098 · doi:10.1214/aop/1176989541 |
| [12] | A. M. Stacey, The existence of an intermediate phase for the contact process on trees,Ann. Probab. 24:1711–1726 (1996). · Zbl 0878.60061 · doi:10.1214/aop/1041903203 |
| [13] | C. C. Wu, The contact process on a tree: behavior near the first phase transition,Stochastic Process. Appl. 57:99–112 (1995). · Zbl 0821.60093 · doi:10.1016/0304-4149(94)00080-D |
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