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

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 \(c_0n^\alpha r^{-\beta}\) 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 \(\alpha>1\), only finitely many cities are founded; (b) if \(\beta<2\alpha\), the share of the largest city tends to \(1\); (c) if \(\alpha>1\) and \(\beta<2\alpha\), only finitely many inhabitants ever live outside the largest city. We prove (a) for all \(c_0>0\), \(\alpha>1\), \(\beta>0\). 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 \(O(t^{2/(\beta+2)})\) almost surely for all parameters; for \(\alpha>1\) the exponent is then lowered in finitely many steps until the founding probabilities are summable. Parts (b) and (c) are false for every \(c_0>0\), \(\alpha>1\) and \(2\alpha/\sqrt3<\beta<2\alpha\): with positive probability two cities near \((\frac12\mp a,\frac12)\) 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 \((2\alpha/\beta)(a+\frac1{12a})\) at the symmetric point, and this slope is less than \(1\) for a suitable spacing exactly when \(2\alpha/\beta<\sqrt3\). Part (b) is also false for every \(\alpha>0\) in the same range of \(\beta\), in particular in the range \(0<\alpha\le1\) of the companion paper of Aldous and Huang. An explicit instance (\(\alpha=2\), \(\beta=3\), any \(c_0\)) is given with all constants. Parts (b) and (c) remain open for \(\beta\le2\alpha/\sqrt3\). 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.