{
  "abstract": "In this paper we prove that a process we call tropical relaxation gives a convergent method for numerically approximating the Monge--Ampère Dirichlet problem on bounded convex planar domains. We approximate the prescribed probability measure by point clouds and construct the least concave tropical roof with zero boundary values and a corner at each point, using repeated updates of affine planes with integral gradients. For universally generic clouds of $N$ points in a fixed compact subset of the interior, convergence of the empirical measures implies uniform convergence of the roofs, divided by $\\sqrt{N}$, to the Aleksandrov solution. The limiting measure may be singular. Euler's and Pick's formulas relate curvature to point counts, with an $O(N^{-1/2})$ discrepancy against compactly supported $C^1$ tests. A removal estimate gives almost-sure uniform convergence for independent samples from any absolutely continuous probability law on the whole domain. For generic clouds in a fixed interior compact set of a rational-slope polygon, suitably rescaled sandpile odometers converge to the same solution along meshes chosen sufficiently fine for each cloud. Their deficit measures converge to its negative Laplacian.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.25878",
  "author_total": 4,
  "authors": [
    {
      "name": "Nikita Kalinin"
    },
    {
      "name": "Ernesto Lupercio"
    },
    {
      "name": "Higinio Serrano"
    },
    {
      "name": "Mikhail Shkolnikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/22p5y3tcbrpd0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.25878",
  "fulltext_url": null,
  "id": "22p5y3tcbrpd0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.25878",
  "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.AG"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
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  "openalex": "W7171659582",
  "pdf_url": "https://arxiv.org/pdf/2607.25878",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/22p5y3tcbrpd0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2607.25878",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.25878"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Many-point tropical relaxation and the Monge--Ampère equation",
  "title_text": "Many-point tropical relaxation and the Monge--Ampère equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/22p5y3tcbrpd0",
  "venue": "arXiv",
  "versions": []
}