{
  "abstract": "We determine explicit bulk and worst-case total variation cutoff profiles for the configuration law of a two-species Togashi--Kaneko model with a fixed population of $n$ particles and weak mutations. Boundary initial conditions introduce an additional logarithmic growth phase and change both the leading mixing-time coefficient and the limiting cutoff profile. Let $κ_1,κ_2>0$ be the autocatalytic rate constants and put $λ_*=κ_1κ_2/(κ_1+κ_2)$. For deterministic initial configurations whose density converges to $m\\in(0,1)$, the total variation distance to equilibrium at time $t_n(s)=\\frac{\\log n}{2λ_*}+\\frac{s}{λ_*}$ converges to $\\mathcal G(r(m)e^{-s})$ for every fixed $s\\in\\mathbb R$, where $\\mathcal G(r)=\\mathrm{d}_{\\mathsf{TV}}(\\mathcal N(r,1),\\mathcal N(0,1))$, $r(m)>0$ is explicit, and $\\mathcal N(a,b^2)$ denotes the Gaussian distribution with mean $a$ and variance $b^2$. For the worst-case distance $\\mathscr D_n(t)$ over all initial configurations, we prove $\\mathscr D_n(T_n+s/λ_*)\\to\\mathcal P(s)$, where $T_n=\\left(\\frac{1}{\\min\\{κ_1,κ_2\\}}+\\frac{1}{2λ_*}\\right)\\log n$.The profile $\\mathcal P$ is the total variation distance between the equilibrium Gaussian and a Gaussian mixture whose random displacement is determined by a gamma-distributed amplitude arising from early growth at the boundary. Unlike the bulk profile, $\\mathcal P$ depends on the mutation parameters. Both cutoffs have windows of constant order, and we identify the asymptotically worst boundary configurations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17775",
  "author_total": 2,
  "authors": [
    {
      "name": "Yangrui Xiang"
    },
    {
      "name": "Wasiur R. KhudaBukhsh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fwphr34avg84t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17775",
  "fulltext_url": null,
  "id": "fwphr34avg84t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17775",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7221818602",
  "pdf_url": "https://arxiv.org/pdf/2610.17775",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fwphr34avg84t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.17775",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17775"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Cutoff profiles and boundary effects in the Togashi-Kaneko model of autocatalytic reactions with weak mutations",
  "title_text": "Cutoff profiles and boundary effects in the Togashi-Kaneko model of autocatalytic reactions with weak mutations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/fwphr34avg84t",
  "venue": "arXiv",
  "versions": []
}