Percolation of the contact process on the regular tree
Abstract
The contact process on the regular tree 𝕋d when d≥ 3 has the two phase transitions of global and of local survival, found by Pemantle and Liggett at values λ1 and λ2. We start the system with every vertex infected and let it relax to what is known as the upper invariant infection. In this stationary state, λp is the critical value beyond which the infected vertices can percolate through 𝕋d, and λpcomplement is the parameter before which the healthy vertices can percolate. We find these are both distinct phase transitions 0<λ1<λp<λ2<λpcomplement<+∞ on 𝕋d when d≥ 7. The most interesting of these comparisons is λ1<λp, which we find for all d≥ 3. This comparison λ1<λp is a long-standing open question on ℤd with d≥ 2 and was not yet found on any other graphs except where λp is infinite.
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BibTeX
@misc{fernley2026percolation,
title = {{Percolation of the contact process on the regular tree}},
author = {John Fernley and Emmanuel Jacob},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09972},
eprint = {2609.09972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09972},
}
RIS
TY - UNPB
TI - Percolation of the contact process on the regular tree
AU - John Fernley
AU - Emmanuel Jacob
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09972
UR - https://arxiv.org/abs/2609.09972
AB - The contact process on the regular tree 𝕋_d when d≥ 3 has the two phase transitions of global and of local survival, found by Pemantle and Liggett at values λ_1 and λ_2. We start the system with every vertex infected and let it relax to what is known as the upper invariant infection. In this stationary state, λ_p is the critical value beyond which the infected vertices can percolate through 𝕋_d, and λ_p^complement is the parameter before which the healthy vertices can percolate. We find these are both distinct phase transitions 0<λ_1<λ_p<λ_2<λ_p^complement<+∞ on 𝕋_d when d≥ 7. The most interesting of these comparisons is λ_1<λ_p, which we find for all d≥ 3. This comparison λ_1<λ_p is a long-standing open question on ℤ^d with d≥ 2 and was not yet found on any other graphs except where λ_p is infinite.
ER -