{
  "abstract": "Lecture notes from a graduate course given by Michael Krivelevich at the School of Mathematical Sciences of Tel Aviv University in the spring semester of 2026. Topics covered include: phase transition and the giant component in $G(n,p)$; long paths and cycles in supercritical and sparse random graphs; thresholds for connectedness and perfect matching; general model of a random subgraph of a finite graph; phase transition and the giant component in the random hypercube; polynomial diameter of the giant component; perfect matchings in the random hypercube.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28552",
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Krivelevich"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9vbhb79ne1xnm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28552",
  "fulltext_url": null,
  "id": "9vbhb79ne1xnm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28552",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.HO"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.28552",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9vbhb79ne1xnm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28552",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28552"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Percolation on Finite Graphs",
  "title_text": "Percolation on Finite Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/9vbhb79ne1xnm",
  "venue": "arXiv",
  "versions": []
}