{
  "abstract": "Let $H$ be a fixed $d$-regular graph with $d\\ge2$, and let $t(H,\\cdot)$ denote its homomorphism density. We study the upper-tail event $t(H,G(n,p))\\ge r^{|E(H)|}$ for fixed $0<p<r<1$ in a dense Erdős--Rényi random graph $G(n,p)$. Near the Lubetzky--Zhao replica-symmetric phase boundary and away from the exceptional target density $(d-1)/d$, we prove that the optimizer of the Chatterjee--Varadhan variational problem on the symmetry-breaking side is bipodal (two-block) and unique up to relabeling. Its block parameters depend analytically on $(p,r)$. To treat the exceptional boundary point, where the nonexceptional theory degenerates, we construct an analytic curve approaching that point from the symmetry-breaking side along which the unique optimizers are nonconstant rank-one bipodal graphons. In both settings, we derive asymptotic expansions of the edge-density deficit and the rate function that governs the exponential decay of the upper-tail probability. Moreover, the conditioned random graph converges in cut distance to the corresponding bipodal optimizer as $n\\to\\infty$. As $(p,r)$ approaches the phase boundary from the symmetry-breaking side, the optimizers converge to their constant limits through two distinct mechanisms. For each fixed nonexceptional target density, one block shrinks to zero measure, giving convergence in $L^1$ but not in $L^\\infty$. Along the exceptional curve, both blocks remain macroscopic: their sizes tend to $1/2$ and all three block densities tend to $(d-1)/d$, yielding convergence in $L^\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15222",
  "author_total": 5,
  "authors": [
    {
      "name": "Sangho Lim"
    },
    {
      "name": "Seonghyuk Im"
    },
    {
      "name": "Taeyoung Kim"
    },
    {
      "name": "Kyeongsik Nam"
    },
    {
      "name": "Hongseok Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/76fv65gzdpr2q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15222",
  "fulltext_url": null,
  "id": "76fv65gzdpr2q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15222",
  "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.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213222266",
  "pdf_url": "https://arxiv.org/pdf/2609.15222",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/76fv65gzdpr2q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15222",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15222"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Bipodal optimizers in the upper-tail variational problem for regular subgraph densities",
  "title_text": "Bipodal optimizers in the upper-tail variational problem for regular subgraph densities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/76fv65gzdpr2q",
  "venue": "arXiv",
  "versions": []
}