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**The contact process on periodic trees.**
*(English)*
Zbl 1434.60295

Electron. Commun. Probab. 25, Paper No. 24, 12 p. (2020); corrigendum ibid. 28, Paper No. 12, 8 p. (2023).

Summary: A little over 25 years ago R. Pemantle [Ann. Probab. 20, No. 4, 2089–2116 (1992; Zbl 0762.60098)] pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values \(\lambda_1\) and \(\lambda_2\) for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generations is \((n,a_1,\ldots, a_k)\) with \(\max_i a_i \le Cn^{1-\delta}\) and \(\log (a_1 \cdots a_k)/\log n \to b\) as \(n\to \infty \). We show that the critical value for local survival is asymptotically \(\sqrt{c (\log n)/n}\) where \(c=(k-b)/2\). This supports Pemantle’s claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

### Citations:

Zbl 0762.60098
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\textit{X. Huang} and \textit{R. Durrett}, Electron. Commun. Probab. 25, Paper No. 24, 12 p. (2020; Zbl 1434.60295)

### References:

[1] | Berger, N., Borgs, C., Chayes, J.T. and Saberi, A. (2005) On the spread o viruses on the internet. Proceedings of the 1CM-SIAM Symposium on Discrete Algorithms, 301-310. · Zbl 1297.68029 |

[2] | Chatterjee, Shirshendu, and Durrett, R. (2009) Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Probab. 37, 2332-2356. · Zbl 1205.60168 |

[3] | Huang, X. and Durret, R. (2020) The contact process on random graphs and Galton-Watson trees. ALEA, to appear. |

[4] | Jiang, Y., Kasssem, R., York G., Zhao, B., Huang, X., Junge, M., and Durrett, R. (2018) The contact process on periodic trees. arXiv:1808.01863. |

[5] | Liggett, T.M. (1999) Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, New York. · Zbl 0949.60006 |

[6] | Pemantle, R. (1992) The contact process on trees. Ann. Probab. 20, 2089-2116. · Zbl 0762.60098 |

[7] | Pemantle, R. · Zbl 1013.60078 |

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