{
  "abstract": "We study a Brownian directed polymer in a centered Gaussian environment that is white in time and colored in space having long-range spatial correlations. For product-type covariances \\(Q(x)\\asymp\\prod_{j=1}^d(1+|x_j|)^{-α_j}\\), with \\(α_j\\in(0,1)\\) and \\(κ=\\sum_jα_j\\), we identify the disorder transition at the marginal value \\(κ=2\\). For \\(κ>2\\), weak disorder holds at sufficiently small inverse temperature; for \\(κ<2\\), the quenched free energy $p(β)$ satisfies \\(-p(β)\\asympβ^{4/(2-κ)}\\) as \\(β\\downarrow0\\). For \\(κ=2\\), strong disorder holds for every \\(β>0\\), while \\(p(β)=0\\) for all sufficiently small \\(β\\), so \\(β_c=0<\\barβ_c\\). We also consider the radial covariance cases, where $ Q(x)\\asymp(1+|x|)^{-\\vartheta}$, when \\(d\\ge3,\\vartheta=2\\) and \\(d=2,\\vartheta\\ge2\\), which was left unanswered in Lacoin~\\cite{Lacoin2011}. When $d=2$, we get \\(\\ln(-p(β))\\asymp-β^{-2}\\) for \\(\\vartheta>2\\) and \\(\\ln(-p(β))\\asymp-β^{-1}\\) for \\(\\vartheta=2\\). The proofs consist of replica coupling, Feynman--Kac variational formula, overlap methods, and continuous-space fractional moments with ordered Wiener-chaos changes of measure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29501",
  "author_total": 3,
  "authors": [
    {
      "name": "Junjie Cao"
    },
    {
      "name": "Guanglin Rang"
    },
    {
      "name": "Jianglun Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fk8va8vjp32c0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29501",
  "fulltext_url": null,
  "id": "fk8va8vjp32c0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29501",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7204937222",
  "pdf_url": "https://arxiv.org/pdf/2608.29501",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fk8va8vjp32c0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29501",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29501"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Disorder Thresholds and Free Energy of Brownian Directed Polymers with Product and Radial Spatial Correlations",
  "title_text": "Disorder Thresholds and Free Energy of Brownian Directed Polymers with Product and Radial Spatial Correlations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/fk8va8vjp32c0",
  "venue": "arXiv",
  "versions": []
}