Unbalanced regimes of the spatial city-growth model

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

At integer time $t\ge1$, cities have distinct positions $x_i\in[0,1]^2$ and populations $N_i(t)\ge1$ summing to $t$. Start with one city of population one at a uniform random position. Fix $c_0,\alpha,\beta>0$, and define \[I(y,t)=c_0\max_i N_i(t)^\alpha\|y-x_i\|_2^{-\beta}.\] An independent uniform point $U\in[0,1]^2$ is selected for immigrant $t+1$. With probability $1/(1+I(U,t))$ the immigrant founds a new city of population one at $U$; otherwise it joins a city attaining the maximum influence at $U$, increasing that city's population by one. Ties may be resolved arbitrarily on the zero-area boundaries. Write $M(t)$ for number of cities, $N_{(1)}(t)=\max_iN_i(t)$, and $N_{(\ge2)}(t)=t-N_{(1)}(t)$. The nondecreasing quantities $M(t)$ and $N_{(\ge2)}(t)$ have limits in $[0,\infty]$.

Statement recorded in the supplied source.

Prove all three assertions: (a) if $\alpha>1$, then $M(\infty)<\infty$ almost surely; (b) if $\beta<2\alpha$, then $N_{(1)}(t)/t\to1$ almost surely; (c) if $\alpha>1$ and $\beta<2\alpha$, then $N_{(\ge2)}(\infty)<\infty$ almost surely. [S2]

Short English statement

Prove eventual finiteness of city formation and dominance of the largest city in the specified unbalanced parameter regimes.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9700023 (AMR-096-0023). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 David J. Aldous, technical notes on the spatial city-growth model, draft dated 25 April 2007, Section 1, model, p. 2; Section 2, Conjecture 1(a)--(c), p. 7. https://www.stat.berkeley.edu/~aldous/Research/OP/cities-notes.pdf.