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

Abstract

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 $n^{-\kappa (H)}\ll p\ll 1$, 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.

Improved the range of sparsity in Theorem 1.1 on upper tail for general H. Also changed order of Thm. 1.1-1.7, new remarks added about other works posted after v3, re-ordering Sec. 5-8 of v3 into Sec. 5-9 here