First-Order Laws for Random Geometric Graphs on the Torus
Abstract
Let GD(n;r) be the random geometric graph generated by n independent uniform points on the D-dimensional torus, with adjacency defined by torus L∞-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⊂(−1/2,1/2)2, the twin count converges to a Poisson variable with mean area(K∘)/16; hence the zero-one law fails for all fixed 0<r<1/2 in the L∞ and Euclidean models. At each critical component threshold nk rnD(k−1)→ a∈(0,∞), the numbers of components of the feasible connected k-vertex types converge jointly to independent Poisson variables, yielding the complete first-order convergence law. Between consecutive thresholds a zero-one law holds. For D≥3, we also construct a definable common-neighborhood configuration of probability order 1/n.
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BibTeX
@misc{haber2026firstorder,
title = {{First-Order Laws for Random Geometric Graphs on the Torus}},
author = {Simi Haber and Mostafa Mirabi and Saharon Shelah},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26230},
eprint = {2609.26230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26230},
}
RIS
TY - UNPB
TI - First-Order Laws for Random Geometric Graphs on the Torus
AU - Simi Haber
AU - Mostafa Mirabi
AU - Saharon Shelah
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26230
UR - https://arxiv.org/abs/2609.26230
AB - 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^∞-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⊂(−1/2,1/2)^2, the twin count converges to a Poisson variable with mean area(K^∘)/16; hence the zero-one law fails for all fixed 0<r<1/2 in the L^∞ and Euclidean models. At each critical component threshold n^k r_n^D(k−1)→ a∈(0,∞), the numbers of components of the feasible connected k-vertex types converge jointly to independent Poisson variables, yielding the complete first-order convergence law. Between consecutive thresholds a zero-one law holds. For D≥3, we also construct a definable common-neighborhood configuration of probability order 1/n.
ER -