Upper tail large deviations of regular subgraph counts in Erdős‐Rényi graphs in the full localized regime

Abstract

For a Δ-regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in G(n,p), an Erdős-Rényi graph on n vertices with edge probability p, has generated significant interests. For p«1 and np^Δ/2»(log n)^1/(vH-2), where vH 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 G(n,p) exceeds its expectation by a constant factor is predicted to hold at a speed n^2pΔlog(1/p), 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 was 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.