Local central limit theorem for general connected subgraph counts in the sparse regime

Establish a local central limit theorem for the number of labelled copies of every fixed connected graph H in the sparse binomial random graph G_{n,p} when p=o(1), under the conditions p\gg n^{-1/m(H)} and n^2(1-p)\gg1, assuming that the corresponding central limit theorem holds.

Background

The paper studies local central limit theorems for subgraph counts X_H in G_{n,p}. Gilmer and Kopparty conjectured that, for connected H, validity of a central limit theorem should imply validity of a local central limit theorem. Previous work had confirmed the conjecture for constant p, while the sparse regime p=o(1) remained unresolved in general.

The present paper proves the conjecture for clique counts H=K_r over essentially the full range n^{-1/m(K_r)}\ll p\le 1/2. Thus, the unresolved problem concerns extending the result from cliques to arbitrary fixed connected graphs in the sparse regime.