The upper-tail exponent conjecture for cliques

Let $H$ be a graph, let $G(n,p)$ be the binomial random graph, and let $d4cd_H$ denote the number of copies of $H$ in $G(n,p)$. Write $v_K$ and $e_K$ for the numbers of vertices and edges of a subgraph $K\subseteq H$, set

$$\Psi(K,n,p)=n^{v_K}p^{e_K},$$

and let $m_H$ and $\Delta_H$ denote the relevant graph-density and maximum-degree parameters. Define

$$M_H(n,p)=\begin{cases} n^2p^{\Delta_H},&p\geq n^{-1/\Delta_H},\\ \displaystyle\min_{K\subseteq H}\bigl(\Psi(K,n,p)\bigr)^{1/\alpha_K^*},&n^{-1/m_H}\leq p\leq n^{-1/\Delta_H}, \end{cases}$$

where $\alpha_K^*$ is the fractional independence number of $K$.

Upper-tail exponent conjecture. For any $H$ and $p>n^{-1/m_H}$,

$$\Pr\left(\xi_H\geq 2{\bf E}\xi_H\right)=\exp\left[-\Theta_H\left(\min\left\{\min_{K\subseteq H,\ e_K>0}\Psi(K,n,p),M_H(n,p)t\right\}\right)\right].$$

Here $t$ is the parameter governing the upper-tail scale in the paper. The conjecture proposes that the true exponent is determined, up to constants depending on $H$, by the largest of the previously established lower-bound mechanisms; the asserted formula remains unproved in general.