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Recent progress in the study of the contact process (see Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly [Ann. Probab. 49 (2021), pp. 244–286]) has verified that the extinction-survival threshold λ 1 \lambda _1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution ξ \xi has an exponential tail. In this paper, we derive the first-order asymptotics of λ 1 \lambda _1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if ξ \xi is appropriately concentrated around its mean, we demonstrate that λ 1 ( ξ ) ∼ 1 / E ξ \lambda _1(\xi ) \sim 1/\mathbb {E} \xi as E ξ → ∞ \mathbb {E}\xi \rightarrow \infty , which matches with the known asymptotics on d d -regular trees. The same results for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well.
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Continuous phase transitions on Galton–Watson trees
Abstract Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $$\mathcal{T}_1$$ be the event that a Galton–Watson tree is infinite and let $$\mathcal{T}_2$$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $$\mathcal{T}_1$$ holds if and only if $$\mathcal{T}_1$$ holds for at least one of the trees initiated by children of the root, and $$\mathcal{T}_2$$ holds if and only if $$\mathcal{T}_2$$ holds for at least two of these trees. The probability of $$\mathcal{T}_1$$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $$\mathcal{T}_2$$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition.
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- Award ID(s):
- 1811952
- PAR ID:
- 10346608
- Date Published:
- Journal Name:
- Combinatorics, Probability and Computing
- Volume:
- 31
- Issue:
- 2
- ISSN:
- 0963-5483
- Page Range / eLocation ID:
- 198 to 228
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0963548321000237
