{
  "abstract": "Mansour, Rastegar and Roitershtein studied the \\emph{roots} of ballistic deposition on a cyclic strip of $K$ sites, namely the sites that receive a particle at height one, and the number $D_{i,K}$ of pairs of consecutive roots at distance $i+1$. They computed the first two moments of $D_{i,K}$ for $i\\le7$. They conjectured that for every $i$ the mean and the variance are exactly linear in $K$, that $D_{i,K}$ satisfies a central limit theorem, and that the standardized vector $(D_{i,K})_{i\\ge1}$ converges to a Gaussian process. We prove these conjectures. The key observation is that the roots are exactly the valleys of a uniformly random circular arrangement, so that $D_{i,K}$ counts occurrences of a family of consecutive patterns. We determine sizes from which the moments are exactly linear; the size for the mean is sharp for every $i$, and those for the second moments cannot be lowered in any case we checked. We give closed forms for the variance constants and for the limiting covariance; the mean constants agree with a known distribution of distances between surface peaks. The reduction and the exact moment formulas are formalized in Lean~4 with Mathlib. We also prove a Wasserstein bound of order $(\\mathbb{E} D_{i,K})^{-1/2}$ that is uniform in $i$, and a local limit theorem. Finally we describe the limit completely when $i$ grows with $K$: it is degenerate, Poisson or Gaussian according as $\\mathbb{E} D_{i,K}$ tends to zero, to a positive constant, or to infinity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21094",
  "author_total": 1,
  "authors": [
    {
      "name": "Mateo Matijasevick"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/944gwc467xjxc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21094",
  "fulltext_url": null,
  "id": "944gwc467xjxc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21094",
  "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": "W7222618352",
  "pdf_url": "https://arxiv.org/pdf/2610.21094",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/944gwc467xjxc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21094",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21094"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Gaps between roots in ballistic deposition on a strip: linear variance, Gaussian limits and a Poisson regime",
  "title_text": "Gaps between roots in ballistic deposition on a strip: linear variance, Gaussian limits and a Poisson regime",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/944gwc467xjxc",
  "venue": "arXiv",
  "versions": []
}