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

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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-096-0023 (Aldous, technical notes on a spatial model of city growth (2007), Conjecture 1).