{
  "abstract": "Understanding how macroscopic wealth distributions emerge from microscopic transaction rules among agents is a central problem in econophysics. While many traditional models restrict agents' wealth to non-negative values, debt is an essential feature of realistic economic systems. In this paper, we introduce a stochastic transaction mechanism into the agent-based model with a central-bank-mediated collective debt limit, as studied in [14]. In our model, each agent probabilistically determines whether to transfer a coin, depending on their current asset or debt level. From a probabilistic perspective, the resulting system can be formulated as a zero-range process with $\\mathbb{Z}$-valued site occupation numbers, where negative values represent debt. We prove that, under an appropriate scaling, the wealth distribution in the stationary state converges to either a two-sided Gamma distribution or a Gamma distribution, depending on the asymptotic behavior of the function governing coin transfers. Our results rigorously identify these previously unconsidered limiting distributions and provide a mathematical framework for understanding how stochastic monetary exchanges give rise to macroscopic wealth distributions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00901",
  "author_total": 1,
  "authors": [
    {
      "name": "Hironobu Sakagawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/chgs4pty6yrpg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00901",
  "fulltext_url": null,
  "id": "chgs4pty6yrpg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00901",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "physics.soc-ph"
    ],
    [
      "openalex",
      "2002|Complex Systems and Time Series Analysis"
    ]
  ],
  "openalex": "W7215850928",
  "pdf_url": "https://arxiv.org/pdf/2610.00901",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/chgs4pty6yrpg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.00901",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00901"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Convergence of stationary distributions for a class of zero-range processes and its application to a wealth distribution model with debt",
  "title_text": "Convergence of stationary distributions for a class of zero-range processes and its application to a wealth distribution model with debt",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/chgs4pty6yrpg",
  "venue": "arXiv",
  "versions": []
}