Large deviations of subgraph counts for sparse Erdős--Rényi graphs

For any fixed simple graph $H=(V,E)$ and any fixed $u>0$, we establish the leading order of the exponential rate function for the probability that the number of copies of $H$ in the Erdős--Rényi graph $G(n,p)$ exceeds its expectation by a factor $1+u$, assuming $n^{-\kappa(H)}\ll p\ll1$, with $\kappa(H) = 1/(2\Delta)$, where $\Delta\ge 1$ is the maximum degree of $H$. This improves on a previous result of Chatterjee and the second author, who obtained $\kappa(H)=c/(\Delta|E|)$ for a constant $c>0$. Moreover, for the case of cycle counts we can take $\kappa$ as large as $1/2$. We additionally obtain the sharp upper tail for Schatten norms of the adjacency matrix, as well as the sharp lower tail for counts of graphs for which Sidorenko's conjecture holds. As a key step, we establish quantitative versions of Szemerédi's regularity lemma and the counting lemma, suitable for the analysis of random graphs in the large deviations regime.