{
  "abstract": "NOTE (October 2026): This paper has been superseded by the joint paper \"Limit shapes for centrally excited walks\", by Ahmed Bou-Rabee, Tuan-Minh Nguyen, Yuval Peres and Arvind Singh (arXiv:2610.10162), which brings together the results and ideas of three independent drafts. We consider a once-excited random walk on $\\mathbb{Z}^d$, $d \\ge 2$, where the walk on its first visit has a bias of constant strength $β>0$ toward the origin and moves like a simple symmetric random walk on subsequent visits to that site. We show that the walk is recurrent and prove an almost sure spherical shape theorem: the trace after $n$ steps is asymptotically a Euclidean ball centered at the origin, with radius proportional to $n^{1/(d+1)}$, and the local times have an asymptotically conical profile. This result confirms a conjecture of Kozma (2007) on the shape of the trace for this model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12138",
  "author_total": 1,
  "authors": [
    {
      "name": "Arvind Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1e9pk4c9kcv5p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12138",
  "fulltext_url": null,
  "id": "1e9pk4c9kcv5p",
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  "landing_url": "https://hal.science/hal-05788575v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
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      "hal",
      "0.math"
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      "openalex",
      "2610|Stochastic processes and statistical mechanics"
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  "openalex": "W7221139390",
  "pdf_url": "https://hal.science/hal-05788575/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1e9pk4c9kcv5p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:52Z",
      "native_id": "hal-05788575",
      "source": "hal",
      "url": "https://hal.science/hal-05788575v1"
    },
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.12138",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12138"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A shape theorem for a radially excited random walk in dimensions d >= 2",
  "title_text": "A shape theorem for a radially excited random walk in dimensions d >= 2",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/1e9pk4c9kcv5p",
  "venue": "HAL",
  "versions": []
}