{
  "abstract": "Finding large lattice subgraphs in sparse Erdős--Rényi random graphs is a classical problem at the interface of random graph theory and algorithms. General bounded-degree embedding and universality theorems give powerful results for broad graph families, but when specialized to square grids they operate at densities substantially larger than the grid-emergence scale. In this paper we exploit the specific geometry of the square grid. We introduce the Quarantined Local Search, a three-phase local algorithm that separates the construction of an initial boundary from the later corner-closure process and controls adaptive negative exposure through bounded pair-test histories. We prove that, for every fixed $δ\\in (0,1)$, there exists $C_δ>0$ such that the algorithm embeds a $k\\times k$ square grid with $k^2\\le (1-δ)n$ in $G(n,p)$ with high probability whenever $p\\ge C_δ\\sqrt{\\ln k/n}$. Thus, for $k^2=Θ(n)$, a density of order $\\sqrt{\\log n/n}$ is sufficient, a factor of order $\\sqrt{\\log n}$ above the corresponding $n^{-1/2}$ emergence scale.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12647",
  "author_total": 2,
  "authors": [
    {
      "name": "Dávid Ferenczi"
    },
    {
      "name": "Alexander Grigoriev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24kt1dzdpfvwh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12647",
  "fulltext_url": null,
  "id": "24kt1dzdpfvwh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12647",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212609595",
  "pdf_url": "https://arxiv.org/pdf/2609.12647",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24kt1dzdpfvwh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12647",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12647"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Local Search for Almost-Spanning Square Grids in Erdős--Rényi Random Graphs",
  "title_text": "Local Search for Almost-Spanning Square Grids in Erdős--Rényi Random Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/24kt1dzdpfvwh",
  "venue": "arXiv",
  "versions": []
}