{
  "abstract": "A centrally excited random walk on $\\mathbb Z^d$ moves like simple random walk, except that its first step from each site other than the origin has a drift of fixed size toward it. Kozma (2007) conjectured that after $n$ steps the visited set approximates a ball with radius of order $n^{1/(d+1)}$. We prove this in every dimension $d\\geq 2$ and show that the rescaled local times converge uniformly to a cone, which is the potential generated by the drift on a ball. The same argument shows that, when the drift is opposite to a subgradient of a norm, the visited set approximates the ball of that norm. In the plane, the outer and inner radii of the visited set differ by at most $n^{1/6}$ times a power of $\\log n$, and the exponent $1/6$ cannot be lowered. Earlier versions of this work appeared in three independent drafts by Nguyen (arXiv:2610.07724), Bou-Rabee and Peres (arxiv:2610.10162v1) and Singh (arxiv:2610.12138). The present paper brings together the results and ideas from these drafts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10162",
  "author_total": 4,
  "authors": [
    {
      "name": "Ahmed Bou-Rabee"
    },
    {
      "name": "Tuan-Minh Nguyen"
    },
    {
      "name": "Yuval Peres"
    },
    {
      "name": "Arvind Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1ystfjrn26myc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10162",
  "fulltext_url": null,
  "id": "1ystfjrn26myc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10162",
  "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": "W7220988606",
  "pdf_url": "https://arxiv.org/pdf/2610.10162",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ystfjrn26myc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.10162",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10162"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Limit shapes for centrally excited walks",
  "title_text": "Limit shapes for centrally excited walks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/1ystfjrn26myc",
  "venue": "arXiv",
  "versions": []
}