{
  "abstract": "The monkey walk is a model of reinforced random walk introduced by Boyer and Solis-Salas as a model for the exploratory behaviour of capuchin monkeys. At each step of the walk, the walker flips a biased coin to determine whether to take a simple, symmetric random walk step on $\\mathbb{Z}^d$ or to \"relocate\", i.e. choose a time uniformly at random in its past and jump to where it was at that time. More recently, Boyer, Falcón-Cortés, Giuggioli and Majumdar looked at a modification of the model, where the probability to relocate is $ξ_0$ when the walker sits at the origin and $ξ>ξ_0$ anywhere else. We prove that, for all $ξ_0\\in (0,1)$, if $ξ$ is sufficiently close to 1, then the monkey walk \"localises\", i.e. the position of the walker at time $n$, $W_n$, converges in distribution without any renormalisation. (We in fact prove convergence of the occupation measure of the walker.) This is in contrast with the case when $ξ= ξ_0$, for which it is known that $W_n/\\sqrt{\\log n}$ converges in distribution to a standard Gaussian (we say that the walk \"diffuses\"). Our proof relies on seeing the monkey walk as an infinitely-many colour Pólya urn (a.k.a. measure-valued Pólya process - MVPP) and then prove convergence of this urn, which is not straightforward as theorems from the literature on MVPPs do not apply directly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21864",
  "author_total": 2,
  "authors": [
    {
      "name": "Wilfred Armfield"
    },
    {
      "name": "Cécile Mailler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d5w18rmyadrky.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21864",
  "fulltext_url": null,
  "id": "d5w18rmyadrky",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21864",
  "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": "W7222578884",
  "pdf_url": "https://arxiv.org/pdf/2610.21864",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d5w18rmyadrky.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21864",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21864"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Localisation of the monkey walk with an \"impurity\" at the origin",
  "title_text": "Localisation of the monkey walk with an \"impurity\" at the origin",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/d5w18rmyadrky",
  "venue": "arXiv",
  "versions": []
}