Year: 2026
Type: preprint
Abstract: Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics of the expected number of adjacent twin pairs in every dimension. In dimension two, more generally, for each origin-symmetric convex connection body $K\subset(-1/2,1/2)^2$, the twin count converges to a Poisson variable with mean $\operatorname{area}(K^\circ)/16$; hence the zero-one law fails for all fixed $0
Source: arXiv (Cornell University)
Authors: Haber, Simi, Mirabi, Mostafa, Shelah, Saharon
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2432433410, G3110727907
OpenAlex ID: W7214079378