{
  "abstract": "We study Bayesian tracking of a diffusing target monitored by a noisy distributed sensor array. Building on an earlier mapping to KPZ growth with a moving defect (or an equivalent directed polymer pinning problem) we determine the phase structure, beyond the previously-studied one-dimensional case, for both Bayes-optimal and suboptimal inference. In $d=2$, theoretical analysis and numerical simulations both give a depinning transition between a successful tracking phase and a failure phase. Weak-coupling RG shows that Bayes-optimal tracking is always successful in $d=2$, but failure can arise from overconfident (suboptimal) inference. In $d=3$, tracking can succeed, or can fail in two distinct ways: the posterior probability distribution may delocalize (no detection), or may become sharply localized, but at the wrong position (a false detection). The two possibilities correspond to Edwards-Wilkinson or Kardar-Parisi-Zhang statistics for the log-posterior. The three phases meet at a Nishimori-like multicritical point on a Bayes-optimal line in a two-parameter phase diagram. (Model misspecification alone can drive depinning into either unpinned phase: underconfidence gives diffuse failure, while overconfidence gives localized-but-wrong failure.) We analyze the transitions between the various phases numerically and with renormalization group arguments. We show that some of these have unusual critical behavior, which the conventional $ε$ expansion fails to describe. Recent rigorous results for directed polymers indicate an alternative scenario. Many of our results, including a scaling relation for exponents at pinning transitions and results for RG flows, are relevant to other phase transitions that involve surface growth or directed polymers in 2+1D or 3+1D.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00144",
  "author_total": 2,
  "authors": [
    {
      "name": "Ewan McCulloch"
    },
    {
      "name": "Adam Nahum"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fy28d7sfen5pp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00144",
  "fulltext_url": null,
  "id": "fy28d7sfen5pp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00144",
  "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",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "cond-mat.soft"
    ],
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "openalex",
      "3104|Theoretical and Computational Physics"
    ]
  ],
  "openalex": "W7206206697",
  "pdf_url": "https://arxiv.org/pdf/2609.00144",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fy28d7sfen5pp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "2609.00144",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00144"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Bayesian Tracking of a Diffusing Target in Two and Three Dimensions",
  "title_text": "Bayesian Tracking of a Diffusing Target in Two and Three Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:00Z",
  "url": "https://research.draconitic.com/paper/fy28d7sfen5pp",
  "venue": "arXiv",
  "versions": []
}