This paper by Jesper Dall and Michael Christensen from the Faculty of Physics at SDU–Odense University explores random geometric graphs (RGGs) in arbitrary dimensions. RGGs are constructed by assigning random coordinates to vertices in a $d$-dimensional box and connecting adjacent points. The authors analyze the critical connectivity, the size of the largest cluster, and the distribution of cluster sizes near the percolation threshold. They find that the critical connectivity $\alpha_c(d)$ approaches the value of 1 in infinite dimensions, but the approach is not linear. The cluster coefficient, a measure of the 'cliquishness' of the graph, is derived and shown to decrease exponentially in higher dimensions, indicating that RGGs are not identical to random graphs even in the limit of infinite dimensionality. The paper also discusses the implications of these findings for graph bi-partitioning and presents an efficient implementation method to study RGGs with up to $4^{11}$ vertices. The results highlight the unique properties of RGGs and their relevance in network theory and percolation theory.This paper by Jesper Dall and Michael Christensen from the Faculty of Physics at SDU–Odense University explores random geometric graphs (RGGs) in arbitrary dimensions. RGGs are constructed by assigning random coordinates to vertices in a $d$-dimensional box and connecting adjacent points. The authors analyze the critical connectivity, the size of the largest cluster, and the distribution of cluster sizes near the percolation threshold. They find that the critical connectivity $\alpha_c(d)$ approaches the value of 1 in infinite dimensions, but the approach is not linear. The cluster coefficient, a measure of the 'cliquishness' of the graph, is derived and shown to decrease exponentially in higher dimensions, indicating that RGGs are not identical to random graphs even in the limit of infinite dimensionality. The paper also discusses the implications of these findings for graph bi-partitioning and presents an efficient implementation method to study RGGs with up to $4^{11}$ vertices. The results highlight the unique properties of RGGs and their relevance in network theory and percolation theory.