Seminar

In the Erdős–Rényi r-uniform hypergraph G(n,p,r), each set of r vertices is included as a hyperedge independently with probability p. The problem of estimating the “infamous upper tail” for the number of copies of a fixed hypergraph H in G(n,p,r) has received considerable attention since it was popularized by Janson and Rucinski, and has been a driving example in the development of Nonlinear Large Deviations Theory. I will discuss new results establishing sharp estimates (large deviation principles) for several classes of H, partially addressing a conjecture of Liu and Zhao. I will also highlight connections with problems and methods from extremal combinatorics, including versions of the regularity and counting lemmas tailored to the large deviations setting. Based on joint works with Amir Dembo, Huy Pham and Nathan Nguyen.