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

TL;DR

This paper determines the exponential decay rate of the probability that subgraph counts in sparse Erdős–Rényi graphs significantly exceed their expected values, improving previous bounds and applying advanced combinatorial tools.

Contribution

It establishes the leading order of large deviation probabilities for subgraph counts in sparse Erdős–Rényi graphs, refining earlier bounds and extending results for cycle counts and Schatten norms.

Findings

Derived the leading order of exponential rate functions for subgraph count deviations.

Improved bounds on the rate function exponent or general graphs.

Established sharp tail bounds for Schatten norms and Sidorenko's conjecture cases.

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 Erd\H{o}s--R\'enyi 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 Szemer\'edi's regularity lemma and the counting lemma, suitable for the analysis of random graphs in the large deviations regime.