The asymptotic enumeration conjecture for clique factors in sparse random graphs

Let $r\geqslant 3$ be constant, let $p\leqslant n^{-2/r+o(1)}$, and let $m=\big\lfloor\binom{n}{r}p^{\binom{r}{2}}\big\rfloor$. Write $F_r(G(n,p))$ for the number of $K_r$-factors in the random graph $G(n,p)$, and let $\Sigma(n,m)$ denote the corresponding benchmark quantity. Clique-factor enumeration conjecture. With high probability,

$$F_r(G(n,p))=e^{-\Theta(n^3/m^2)}\,\Sigma(n,m)\cdot\mathbf{1}\big[F_r(G(n,p))>0\big].$$

At $p=\Theta(q_r(n))$, the correction term has the form $\exp\big(-\Theta(n/(\log n)^2)\big)$ and reflects variation in the number of intersections between copies of $K_r$. The authors expect the statement to hold up to roughly $m=n^{4/3}$, where variation in the number of copies of $K_r$ has a larger effect.