{
  "abstract": "We study the limit shape of the range of the random walk excited to the center on $\\mathbb Z^d$, with $d\\geq2$. A cookie is placed at each nonzero vertex. On its first visit to such a vertex, the walker consumes the cookie and takes its next nearest-neighbor step with a bias toward the origin. On a visit to the origin, or on later visits to a nonzero vertex, no cookie is available and the walker moves to a neighbor uniformly at random. We prove that the set of vertices visited up to time $n$, rescaled by $n^{-1/(d+1)}$, converges in probability in Hausdorff distance to an explicit $\\ell^1$ ball. This answers a question raised by Kozma (2007).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07724",
  "author_total": 1,
  "authors": [
    {
      "name": "Tuan-Minh Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6vtzd3mdfkntz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07724",
  "fulltext_url": null,
  "id": "6vtzd3mdfkntz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07724",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
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  ],
  "openalex": "W7221061258",
  "pdf_url": "https://arxiv.org/pdf/2610.07724",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6vtzd3mdfkntz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2610.07724",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07724"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Shape theorem for excited random walks on the integer lattice",
  "title_text": "Shape theorem for excited random walks on the integer lattice",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/6vtzd3mdfkntz",
  "venue": "arXiv",
  "versions": []
}