{
  "abstract": "The contact process on the regular tree $\\mathbb{T}_d$ when $d\\geq 3$ 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 $\\mathbb{T}_d$, 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}<+\\infty\\] on $\\mathbb{T}_d$ when $d\\geq 7$. The most interesting of these comparisons is $λ_1<λ_p$, which we find for all $d\\geq 3$. This comparison $λ_1<λ_p$ is a long-standing open question on $\\mathbb{Z}^d$ with $d\\geq 2$ and was not yet found on any other graphs except where $λ_p$ is infinite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09972",
  "author_total": 2,
  "authors": [
    {
      "name": "John Fernley"
    },
    {
      "name": "Emmanuel Jacob"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7vsxcyr8p91dd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09972",
  "fulltext_url": null,
  "id": "7vsxcyr8p91dd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09972",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7212138448",
  "pdf_url": "https://arxiv.org/pdf/2609.09972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7vsxcyr8p91dd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09972"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Percolation of the contact process on the regular tree",
  "title_text": "Percolation of the contact process on the regular tree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/7vsxcyr8p91dd",
  "venue": "arXiv",
  "versions": []
}