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 $n^{-\kappa(H)}\ll p\ll1$, 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2018
- DOI:
- 10.48550/arXiv.1809.11148
- arXiv:
- arXiv:1809.11148
- Bibcode:
- 2018arXiv180911148C
- Keywords:
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- Mathematics - Probability;
- Mathematics - Combinatorics;
- 60F10;
- 05C80;
- 60C05;
- 60B20
- E-Print:
- 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