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Local limit theorem for joint subgraph counts
Ashwin Sah, Mehtaab Sawhney, Daniel G. Zhu · 2024-12-13 · via math.PR updates on arXiv.org

Extending a previous result of the first two authors, we prove a local limit theorem for the joint distribution of subgraph counts in the Erdős-Rényi random graph $G(n,p)$. This limit 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. As an application, we show a number of results concerning the existence and enumeration of proportional graphs and related concepts, answering various questions of Janson and collaborators in the affirmative.

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