{"paper":{"title":"Phase transition and critical behavior in hierarchical integer-valued Gaussian and Coulomb gas models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Haiyu Huang, Marek Biskup","submitted_at":"2024-12-12T05:54:01Z","abstract_excerpt":"Given a square box $\\Lambda_n\\subseteq\\mathbb Z^2$ of side length $L^n$ with $L,n>1$, we study hierarchical random fields $\\{\\phi_x\\colon x\\in\\Lambda_n\\}$ with law proportional to ${\\rm e}^{\\frac12\\beta(\\phi,\\Delta_n\\phi)}\\prod_{x\\in\\Lambda_n}\\nu({\\rm d}\\phi_x)$, where $\\beta>0$ is the inverse temperature, $\\Delta_n$ is a hierarchical Laplacian on $\\Lambda_n$, and $\\nu$ is a non-degenerate $1$-periodic measure on $\\mathbb R$. Our setting includes the integer-valued Gaussian field (a.k.a. DG model or Villain Coulomb gas) and the sine-Gordon model. Relying on renormalization group analysis we de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.08964","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.08964/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}