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- W2893478813 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--Renyi graph $G(n,p)$ exceeds its expectation by a factor $1+u$, assuming $n^{-kappa(H)}ll pll1$, with $kappa(H) = 1/(2Delta)$, where $Deltage 1$ 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 Szemeredi's regularity lemma and the counting lemma, suitable for the analysis of random graphs in the large deviations regime." @default.
- W2893478813 created "2018-10-05" @default.
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- W2893478813 date "2018-09-28" @default.
- W2893478813 modified "2023-09-27" @default.
- W2893478813 title "Large deviations of subgraph counts for sparse ErdH{o}s--R'enyi graphs" @default.
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