Anti-concentration for subgraph counts in random graphs

Fix a graph H and some p∈(0,1), and let XH be the number of copies of H in a random graph G(n,p). Random variables of this form have been intensively studied since the foundational work of Erdős and Renyi. There has been a great deal of progress over the years on the large-scale behaviour of XH, 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 XH 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 x∈N we have Pr(XH=x)≤n1−v(H)+o(1). Our proof proceeds by iteratively breaking XH into different components which fluctuate at “different scales”, and relies on a new anti-concentration inequality for random vectors that behave “almost linearly.”