Upper Tail Large Deviations of Regular Subgraph Counts in Erdős-Rényi Graphs in the Full Localized Regime

For a -regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in , an Erdős-Rényi graph on n vertices with edge probability p, has generated significant interest. For and , where is the number of vertices in H, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of H in exceeds its expectation by a constant factor is predicted to hold at a speed , and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years, covering progressively broader ranges of p, the upper tail large deviations for cliques of fixed size were proved by Harel, Mousset, and Samotij in the entire localized regime. This paper establishes the conjecture for all connected regular graphs in the whole localized regime. © 2021 Wiley Periodicals LLC.