{
  "schema_version": 1,
  "problem_number": "AMR-096-0023",
  "title": "On Conjecture 1 of Aldous for a Spatial Model of City Growth: A Proof of Part (a), and Counterexamples to Parts (b) and (c) for 2α/√3 < β < 2α",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In rough notes of 2007 on a spatial model of city growth, D. J. Aldous stated a three-part conjecture about the unbalanced regime. Immigrants arrive one at a time at uniform random points of the unit square; a city of population n exerts the influence c0 n^α r^(−β) at distance r; an immigrant founds a new city with probability 1/(1+I), where I is the largest influence at its position, and otherwise joins the city whose influence there is largest. The conjecture asserts, almost surely: (a) if α > 1, only finitely many cities are founded; (b) if β < 2α, the share of the largest city tends to 1; (c) if α > 1 and β < 2α, only finitely many inhabitants ever live outside the largest city. We prove (a) for all c0 > 0, α > 1, β > 0. A packing bound for the cities founded where the influence is small, combined with Ville's maximal inequality, shows that the number of cities is O(t^(2/(β+2))) almost surely for all parameters; for α > 1 the exponent is then lowered in finitely many steps until the founding probabilities are summable. Parts (b) and (c) are false for every c0 > 0, α > 1 and 2α/√3 < β < 2α: with positive probability two cities near (1/2 ∓ a, 1/2) share the population in nearly equal parts forever. The reason is that the two-city share map, the area of a city's sphere of influence as a function of its share of the population, has the slope (2α/β)(a + 1/(12a)) at the symmetric point, and this slope is less than 1 for a suitable spacing exactly when 2α/β < √3. Part (b) is also false for every α > 0 in the same range of β, in particular in the range 0 < α ≤ 1 of the companion paper of Aldous and Huang. An explicit instance (α = 2, β = 3, any c0) is given with all constants. Parts (b) and (c) remain open for β ≤ 2α/√3. The model and the conjecture are Aldous's, and the tools (Ville's and Doob's inequalities, circles of Apollonius) are standard. To our knowledge the results are new; no answer to the conjecture was found in the literature accessible to us, and a search that finds nothing is not a proof of novelty. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.PR"
  ],
  "keywords": [
    "spatial model of city growth",
    "sphere of influence",
    "weighted Voronoi tessellation",
    "unbalanced growth",
    "random processes with reinforcement",
    "generalized Pólya urn",
    "Ville's inequality",
    "circle of Apollonius",
    "part (a) proved",
    "counterexamples to parts (b) and (c) for 2α/√3 < β < 2α",
    "Aldous open problems",
    "UnsolvedMath",
    "AMR-096-0023",
    "math.PR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-14",
  "publication_date": "2026-10-14",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-14",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-096-0023/",
  "pdf_url": "https://eulersolve.org/papers/amr-096-0023/paper.pdf?v=319a8f49e0c4",
  "doi": "10.5281/zenodo.23367464",
  "zenodo_record_url": "https://zenodo.org/records/23367464",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Conjecture 1 of D. J. Aldous's technical notes on a spatial model of city growth (2007) per part: (a) is proved for every c0 > 0, α > 1, β > 0; (b) and (c) are false for every c0 > 0, α > 1 and 2α/√3 < β < 2α, and (b) is false for every α > 0 in that range of β; (b) and (c) remain open for β ≤ 2α/√3. The model and the conjecture are Aldous's; the tools (Ville's and Doob's inequalities, circles of Apollonius) are standard. Novelty is not certified.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}