{
  "abstract": "Let $z\\in \\mathbb{H}:=\\{z\\in\\mathbb{C}:\\Im(z)>0\\}$ and $Λ:=\\frac{1}{\\sqrt{\\Im(z)}}\\big({\\mathbb Z}\\oplus z{\\mathbb Z}\\big)$, and set $ θ(α;z):=θ(α;Λ)=\\sum_{(m,n)\\in\\mathbb{Z}^2}e^{-πα\\frac{|m+nz|^2}{\\Im(z)}}. $ We study the minimization of the functional $θ(α;z)-bθ(2α;z)$ and establish a complete phase diagram for all $α>0$ and $b\\in(0,\\sqrt{2})$. For $0<b<\\sqrt{2}$, an explicit threshold $b_1=0.0767767081\\cdots\\in(0,1/10)$ separates two distinct types of phase transitions. Type I: $0<b<b_1$: it admits a type of ``hexagonal--rhombic--square--rectangular'' phase transition. Type II: $b_1\\le b<\\sqrt{2}$: it admits a type of ``hexagonal--square--rectangular'' phase transition, without passing through the rhombic lattices. These phase transitions contrast with Montgomery's theorem \\cite{Mon1988}, where the hexagonal lattice minimizes the single theta function at all scales, and the recent Bétermin-Faulhuber-Steinerberger theorem \\cite{BFS2026} for translated Gaussian lattice sums, where the hexagonal lattice prevails in a min-max structure. In our setting, the density parameter $α$ governs the sequence of optimal lattice shapes, while the competition parameter $b$ controls the emergence of the rhombic phase, which exists for $b<b_1$ and vanishes for $b\\ge b_1$. As an application, we resolve Bétermin's conjecture \\cite{Betermin2021AHP}. Bétermin predicted a Type I phase transition for the functional $θ(α;z)-(1/10)θ(2α;z)$ with $b=1/10$. We instead rigorously establish that no rhombic lattice is a global minimizer at any density and prove that the system undergoes a Type II phase transition, thereby settling the conjecture in corrected form.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35437",
  "author_total": 2,
  "authors": [
    {
      "name": "Kaixin Deng"
    },
    {
      "name": "Senping Luo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ehdkhkhgaqcb3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35437",
  "fulltext_url": null,
  "id": "ehdkhkhgaqcb3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35437",
  "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.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223951572",
  "pdf_url": "https://arxiv.org/pdf/2610.35437",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ehdkhkhgaqcb3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.35437",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35437"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The role of density in phase transitions and new perspectives on theta functions",
  "title_text": "The role of density in phase transitions and new perspectives on theta functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/ehdkhkhgaqcb3",
  "venue": "arXiv",
  "versions": []
}