×

zbMATH — the first resource for mathematics

The contact process on periodic trees. (English) Zbl 1434.60295
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
PDF BibTeX XML Cite
Full Text: DOI Euclid
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.