{
  "abstract": "This paper studies the last-coverage vertex of a nearest-neighbour random walk on a cycle: the vertex most likely to be the final one discovered. We show that the visited vertices always form a single contiguous arc of the ring, a property we call the Connected Arc Property. This lets the walk be replaced by a two-state time-inhomogeneous Markov chain, which we call the interval-expansion representation, in which each expansion step has a probability given by the classical gambler's ruin formula. Within this framework we prove that the last-coverage probability of node k in an n-node cycle is (1-r)r^(n-1-k)/(1-r^(n-1)), where r = (1-p)/p and p is the clockwise transition probability. As p approaches 1/2, we recover the uniform distribution, giving an independent derivation of a classical result. For a walk with a deterministic time-varying bias, we give an exact series representation for the last-coverage probabilities, together with a finite truncation that serves as a computable, error-bounded approximation. All results are validated by Monte Carlo simulation, and we run experiments on sinusoidal time-varying bias. Finally, we compare generalized Petersen graphs with Erdos-Renyi graphs of matching size, finding that structural symmetry is associated with more uniform last-coverage distributions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15115",
  "author_total": 2,
  "authors": [
    {
      "name": "Kiyan Idnani"
    },
    {
      "name": "Jad Salem"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dy962p3re6kn2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15115",
  "fulltext_url": null,
  "id": "dy962p3re6kn2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15115",
  "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": "W7221778480",
  "pdf_url": "https://arxiv.org/pdf/2610.15115",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dy962p3re6kn2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15115",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15115"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Last-Coverage Probabilities of Random Walks on Cycle Graphs for Symmetric, Biased, and Time-Dependent Walks",
  "title_text": "Last-Coverage Probabilities of Random Walks on Cycle Graphs for Symmetric, Biased, and Time-Dependent Walks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/dy962p3re6kn2",
  "venue": "arXiv",
  "versions": []
}