{
  "abstract": "It is conjectured that in the Lorentz mirror model on $\\Z^2$ all trajectories are finite. Quas showed that, almost surely, the two halves of any infinite trajectory, cut at an edge, meet at infinitely many vertices, each of which is pivotal: changing its state can make the trajectory finite. We construct plane graphs on which infinite trajectories exist, and all of them have this property. Given $p\\in(0,1)$, we replace each edge of a random one-ended subtree of $\\Z^2$ by a number of parallel edges, depending on $p$, that is at most logarithmic in the height of the corresponding descendant tree. The medial graph of the resulting plane multigraph is a factor of the tree, and for the uniform spanning tree it has finite vertex intensity. In the mirror model on this medial graph, with probability $p$ for mirrors along primal edges and any probability $r\\in(0,1-p]$ for mirrors along dual edges, infinite trajectories exist, and the two halves of each of them meet infinitely often. The construction rests on Bernoulli percolation: every random one-ended locally finite tree can be decorated in this way such that bond percolation with parameter $p$ is critical and percolates, and every vertex of the infinite cluster has infinitely many pivotal edges. We characterize this property for bond and site percolation and for mirrors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12460",
  "author_total": 1,
  "authors": [
    {
      "name": "Martin P. W. Zerner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3a583a64sa2mk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12460",
  "fulltext_url": null,
  "id": "3a583a64sa2mk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12460",
  "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": "W7221194765",
  "pdf_url": "https://arxiv.org/pdf/2610.12460",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3a583a64sa2mk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12460",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12460"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Infinite light rays and infinite clusters with infinitely many pivots",
  "title_text": "Infinite light rays and infinite clusters with infinitely many pivots",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/3a583a64sa2mk",
  "venue": "arXiv",
  "versions": []
}