Scaling of components in critical long-range geometric random graphs on the 2-dim torus

We consider random graphs on the set of N 2 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 ( N ρ ) − 1 . This is a versatile example of an inhomogeneous random graph that is not of rank 1. Here, we study the critical phase: the main result is the weak limit of the size of the largest connected component rescaled with ( N 2 ) − 2 / 3 described by a diffusion process. This completes the proof that in all regimes, the model is within the same universality class as the Erdős-Rényi graph.