Upper tail bounds for cycles

Abstract

This paper examines bounds on upper tails for cycle counts in $G_{n,p}$. For a fixed graph $H$ define $\xi_H= \xi_H^{n,p}$ to be the number of copies of $H$ in $G_{n,p}$. It is a much studied and surprisingly difficult problem to understand the upper tail of the distribution of $\xi_H$, for example, to estimate \begin{equation*} \mathbb{P}(\xi_H > 2 \mathbb{E}\xi_H). \end{equation*} The best known result for general $H$ and $p$ is due to Janson, Oleszkiewicz, and Ruci\'nski, who, in 2004, proved \begin{align}\label{a:JOR} \exp[-O_{H, \eta}(M_H(n,p) \ln(1/p))]&<\mathbb{P}(\xi_H > (1+\eta)\mathbb{E} \xi_H)\\&<\exp[-\Omega_{H, \eta}(M_{H}(n,p))].\nonumber \end{align} Thus they determined the upper tail up to a factor of $\ln(1/p)$ in the exponent. There has since been substantial work to improve these bounds for particular $H$ and $p$. We close the $\ln(1/p)$ gap for cycles, up to a constant in the exponent. Here the lower bound given by JOR is the truth for $l$-cycles when $p> \frac{\ln^{1/(l-2)}n}{n}$.