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For any fixed simple graph $H=(V,E)$ and any fixed $u>0$, we establish theleading order of the exponential rate function for the probability that thenumber of copies of $H$ in the Erd\H{o}s--R\'enyi graph $G(n,p)$ exceeds itsexpectation 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, whoobtained $\kappa(H)=c/(\Delta|E|)$ for a constant $c>0$. Moreover, for the caseof cycle counts we can take $\kappa$ as large as $1/2$. We additionally obtainthe sharp upper tail for Schatten norms of the adjacency matrix, as well as thesharp lower tail for counts of graphs for which Sidorenko's conjecture holds.As a key step, we establish quantitative versions of Szemer\'edi's regularitylemma and the counting lemma, suitable for the analysis of random graphs in thelarge deviations regime.

Large deviations of subgraph counts for sparse Erd\H{o}s--R\'enyi graphs