{
  "abstract": "This paper studies a class of \\textbf{exponentially age-preferential random growth graphs} $G(m,β)$: at each time $n$ a new vertex $v_n$ is added, and $m$ edges are drawn to existing vertices independently, with the probability that a target $v_k$ is chosen proportional to $e^{βk}$ ($β>0$), so that the graph's growth strongly favors ``young'' vertices. We systematically investigate two fundamental probabilistic problems of this model and supply the proof details: (1)~\\textbf{Recurrence}: when $β>0$, the simple random walk on $G$ is recurrent with probability $1$ (Theorem~\\ref{thm:rec}); (2)~\\textbf{Infinite collisions}: for every $β>0$, two independent simple random walks on $G$ meet infinitely often with probability $1$ (Theorem~\\ref{thm:icp}). The core of the proof is the following: by the Borel--Cantelli lemma combined with the \\textbf{memoryless property} of exponential weights, $G$ is almost surely ``sausage-like'' --- there exist infinitely many bottlenecks consisting of exactly $m$ parallel edges; the cascade of bottlenecks makes the effective resistance diverge, yielding recurrence; the renewal process $\\{T_k\\}$ constructed at the bottleneck times has i.i.d.\\ stationary increments with finite moments of every order for the first-passage times, and combined with a moderate-deviation estimate we verify Peres' Green-function criterion for infinite collisions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02329",
  "author_total": 3,
  "authors": [
    {
      "name": "Jia-shan Tang"
    },
    {
      "name": "Feng Wang"
    },
    {
      "name": "Xian-Yuan Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/32y1m3qthfx5y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02329",
  "fulltext_url": null,
  "id": "32y1m3qthfx5y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02329",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02329",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/32y1m3qthfx5y.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02329",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02329"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Random Walk on age-preferential random growth processes",
  "title_text": "Random Walk on age-preferential random growth processes",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/32y1m3qthfx5y",
  "venue": "arXiv",
  "versions": []
}