Sparse-regime anti-concentration for general connected subgraph counts

Determine the anti-concentration behaviour of the number X_H of labelled copies of an arbitrary fixed connected graph H in G_{n,p} when p=o(1), including the asymptotic order of \sup_x \mathbb{P}(X_H=x).

Background

For a discrete random variable, anti-concentration concerns the largest point probability, \sup_x \mathbb{P}(X=x). An LCLT gives an asymptotically sharp description of this quantity for subgraph counts.

Prior work had obtained asymptotically optimal anti-concentration results for connected graphs when p is constant, and the triangle case was known in the sparse regime. The paper's clique LCLT supplies the corresponding asymptotic answer for K_r, but the general connected-graph case in the sparse regime remains unresolved.