Largest-city growth at alpha=1

Definitions and mathematical statement

In the source's city-growth model, let $N_{(1)}(t)$ denote the largest city population at model time $t$, as described by the supplied summary. Fix parameters $\alpha=1$ and $\beta>2$. Conjecture 22 asserts \[ N_{(1)}(t)=t(\log t)^{1-2/\beta+o(1)}\qquad\text{almost surely as }t\to\infty. \] Precisely, with $\gamma=1-2/\beta$, the logarithmic correction statement is \[ \Pr\left(\lim_{t\to\infty} \frac{\log(N_{(1)}(t)/t)}{\log\log t}=\gamma\right)=1, \] where the limit follows the model's time index and populations are positive for the relevant large times. Equivalently, with probability one, for every $\varepsilon>0$, there is a finite random $t_0$ such that \[ t(\log t)^{\gamma-\varepsilon}\leq N_{(1)}(t) \leq t(\log t)^{\gamma+\varepsilon}\qquad(t\geq t_0). \] The $o(1)$ is in the logarithmic exponent and may depend on the sample path. No ratio-one asymptotic with a specified multiplicative constant is asserted.

The underlying Aldous--Huang model's probability law, city creation and population dynamics, meaning of $\alpha,\beta$, and time normalization are not included in the excerpt. These must be supplied to define $N_{(1)}$ and its almost-sure event. The statement is for each fixed admissible $\beta>2$; no uniformity or shared probability space over all $\beta$ is added.

Short English statement

At $\alpha=1$, the largest city is conjectured to grow almost surely like $t$ times a logarithmic factor whose exponent tends to $1-2/\beta$, for fixed $\beta>2$. The exact city model and time scale are missing.

Sources

• Local UnsolvedMath registry, permanent ID 9700025, source code catalogue entry; David Aldous; no proposal year supplied; updated 2026-08-06.
• Aldous, Open problems, cities.html and technical notes, Conjecture 22: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html. Registry access date 2026-07-29; HTML text with linked primary-PDF text. Background reports 23 active pages, with delegated explicit subproblems separated and compound bullets retained. Present status and redistribution terms require review.
• The supplied summary identifies the Aldous--Huang model and cites arXiv:1209.5120, reporting that the $\alpha=1$ case remains unsettled. No title or exact later reference is supplied; these status and search claims were not independently verified here.