Scaling of Components in Critical Geometric Random Graphs on 2-dim Torus

2023/08/15 by Goriachkin, Vasilii, Turova, Tatyana
#05C80 #60G42 #60G50 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2308.07696

Abstract

We consider random graphs on the set of N² vertices placed on the discrete 2-dimensional torus. The edges between pairs of vertices are independent, and their probabilities decay with the distance ρ between these vertices as (Nρ)^-1. This is an example of an inhomogeneous random graph which is not of rank 1. The reported previously results on the sub- and super-critical cases of this model exhibit great similarity to the classical Erdős-Rényi graphs. Here we study the critical phase. A diffusion approximation for the size of the largest connected component rescaled with (N²)^-2/3 is derived. This completes the proof that in all regimes the model is within the same class as Erdős-Rényi graph with respect to scaling of the largest component.

Related

• Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph
• Phase transition in random distance graphs on the torus
• Component sizes for large quantum erdos renyi graph near criticality
• An elementary approach to component sizes in critical random graphs
• Critical scaling limits of the random intersection graph