Tatyana Turova: Scaling for the Largest Component in Critical Long-Range Geometric Random Graphs

Abstract: We consider random graphs on the set of vertices placed on the discrete torus. The edges between pairs of vertices are independent, and their probabilities decay polynomial of degree $\alpha$ with the distance between these vertices. The question of interest is the phase transition of the size of the largest connected component. The off-critical phases are known, but the critical phase was studied only for the particular case of $\alpha$ when the model falls into the same universality class as the Erdős-Rényi graph. We extend the analysis of the critical phase beyond this phase, covering a wide range of the parameter $\alpha$. The weak limit of the rescaled size of the largest connected component is described by a diffusion process whose coefficients depend on the number of triangles in the graph.