{
  "abstract": "We show that the vacuum partition function of the $q$-deformed oscillator \\[ L_q(z):= \\langle0|e^{zT_q}|0\\rangle, \\qquad T_q := \\mathsf{a}_q + \\mathsf{a}_q^* \\] satisfies the Lee-Yang property, i.e., all zeros of $L_q$ lie on the imaginary line for all $q\\in[-1, 1)$, proving the conjectures of Okuyama'23. Using the connection between $q$-deformed oscillators and the Sachdev-Ye-Kitaev (SYK) Hamiltonian in the doubly-scaled regime, we show that the annealed SYK partition function satisfies the Lee-Yang property in the thermodynamic limit, that is, all of its Fisher zeros are on the imaginary axis. As a result, we show that the quenched SYK partition function admits a zero-free strip of constant width with high probability, rigorously complementing the large$-k$ replica calculations of Zlokapa and Kiani'26. We show the imaginary-rootedness of $L_q$ by expressing $φ_q(z):= L_q(\\mathbf{i} z)$ as a locally uniform limit of ``$q$-deformed'' matching polynomials and then showing that those $q$-deformed polynomials are real-rooted. This establishes that $φ_q$ is real-rooted, and in fact belongs to the \\emph{Laguerre-Pólya} class. The real-rootedness of the $q$-deformed polynomials yields novel log-concavity relations on the \\emph{$q$-deformed matching moments} \\[M_\\ell(q):= \\sum_{π\\in\\mathcal{P}_2(2\\ell)}q^{\\operatorname{cr}(π)}.\\] We show the real-rootedness of the $q$-deformed polynomials by using the Touchard--Riordan formula to derive a partial differential equation relating their $q$-derivatives to their $z$-derivatives, and then analyzing the resulting motion of their roots.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03624",
  "author_total": 2,
  "authors": [
    {
      "name": "Arpon Basu"
    },
    {
      "name": "Siddhant Midha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0n5fcj5qms860.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03624",
  "fulltext_url": null,
  "id": "0n5fcj5qms860",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03624",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03624",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0n5fcj5qms860.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:04Z",
      "native_id": "2611.03624",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03624"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Lee-Yang theorems for $q$-deformed oscillators and doubly-scaled SYK",
  "title_text": "Lee-Yang theorems for q-deformed oscillators and doubly-scaled SYK",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/0n5fcj5qms860",
  "venue": "arXiv",
  "versions": []
}