Recently, Sah and Sawhney (2022) showed that for a connected graph H, the number of copies of H inside the Erdős-Rényi random graph G(n, p) obeys a local central limit theorem, meaning that the probability that the number of copies equals a specific integer approaches "what one might expect from a normal distribution" as n grows to infinity. In this talk, we discuss an extension of this result to the joint distribution of subgraph counts. The limit here can be described as a nonlinear transformation of a multivariate normal distribution, where the components of the multivariate normal correspond to the "graph factors" of Janson. We also discuss an application to the existence and enumeration of proportional graphs and related concepts.

Joint work with Ashwin Sah and Mehtaab Sawhney.