Scaling of Components in Critical Geometric Random Graphs on 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 $\rho$ between these vertices as $(N\rho {)}^{-1}$. 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 $(N^2{)}^{-2\mathrm{/}3}$ 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.