Anti-concentration for subgraph counts in random graphs

Abstract

Fix a graph H and some $\mathit{p}\in (0,1)$, and let ${\mathit{X}}_{\mathit{H}}$ be the number of copies of H in a random graph $\mathbb{G}(\mathit{n},\mathit{p})$. Random variables of this form have been intensively studied since the foundational work of Erdős and Rényi. There has been a great deal of progress over the years on the large-scale behaviour of ${\mathit{X}}_{\mathit{H}}$, but the more challenging problem of understanding the small-ball probabilities has remained poorly understood until now. More precisely, how likely can it be that ${\mathit{X}}_{\mathit{H}}$ falls in some small interval or is equal to some particular value? In this paper, we prove the almost-optimal result that if H is connected then for any $\mathit{x}\in \mathbb{N}$ we have $Pr({\mathit{X}}_{\mathit{H}}=\mathit{x})\le {\mathit{n}}^{1-\mathit{v}(\mathit{H})+\mathit{o}(1)}$. Our proof proceeds by iteratively breaking ${\mathit{X}}_{\mathit{H}}$ into different components which fluctuate at “different scales”, and relies on a new anti-concentration inequality for random vectors that behave “almost linearly.”