{
  "abstract": "A Markov chain with exit rates $q_i$ runs for $N$ steps with transition matrix $I+(T/N)Q$. On each irreducible face that carries start mass, we prove a local limit theorem for the occupation counts with an explicit constant and relative error $O(1/T)$, uniformly for $1\\le T\\le\\varepsilon N$ with $\\varepsilon\\max_iq_i<1$ and on compact subsets of the open face. Up to $O(1)$, minus the log-probability of counts $n$ with support $F$ is $|F|-1$ times $\\log(N/\\sqrt T)$ plus $N$ times the Donsker and Varadhan functional of $I+(T/N)Q$ at $n/N$. Every admissible face has an explicit power of $T$ in its largest probability for large $T\\le N^{1/3}$, and a constant given by a finite-dimensional variational formula. For $T=τ\\log N$ the face of the mode minimizes $τλ_F+|F|-1$ at first order, with $λ_F$ the exit rate. If an admissible face has $Δ\\ge1$ more states than another and an exit rate smaller by $δ>0$, we locate every crossing of their largest probabilities for large $T\\le N^{1/3}$ up to $O(1/\\log N)$. For two irreducible faces the crossing is unique for large $N$ in $1\\le T\\le\\varepsilon N$, with coefficient $1/2$ of $\\log\\log N$. Near a facet, the continuous-time occupation density is the facet density times an explicit kernel, with relative error $O(1/T)$ at times $x\\le cT$ off the facet, $c$ small. The counts obey a lattice form with extra error $T^2/N$ for $T\\le\\varepsilon_1\\sqrt N$. The kernel is Bessel for $x=o(T)$ at facet profiles where the removed state has equal entry and exit laws, and at a profile where the rate has a first correction, of order $m\\ge3$, if and only if $x=o(T^{1-2/m})$. On the complete graph with the uniform start, each face of at least 2 states crosses a vertex exactly once.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09258",
  "author_total": 1,
  "authors": [
    {
      "name": "Arjun Pemmasani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/766q8kzqxnj1d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09258",
  "fulltext_url": null,
  "id": "766q8kzqxnj1d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09258",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7220989379",
  "pdf_url": "https://arxiv.org/pdf/2610.09258",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/766q8kzqxnj1d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.09258",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09258"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Occupation counts of rarely switching Markov chains: stratified local laws, support selection and the boundary layer",
  "title_text": "Occupation counts of rarely switching Markov chains: stratified local laws, support selection and the boundary layer",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/766q8kzqxnj1d",
  "venue": "arXiv",
  "versions": []
}