Mean-field approximation conjecture for upper tails of strictly balanced subgraph counts

Let $H$ be a strictly balanced graph, with $v_H$ vertices, $e_H$ edges, maximum degree $\Delta$, and parameter $\alpha_H^*$. Let $\Psi_{p,H}(\delta)$ denote the mean-field variational problem for the upper tail of the number $N(H,\mathbb{G}(n,p))$ of copies of $H$ in the Erdős–Rényi random graph. Mean-field approximation conjecture. For every $\varepsilon\in(0,1)$, $\delta>0$, and $p$ satisfying

$$p\ll n^{-1/\Delta},\qquad n^{v_H}p^{e_H}\gg (\log n)^{\alpha_H^*/(\alpha_H^*-1)},$$

for all sufficiently large $n$,

$$(1-\varepsilon)\Psi_{p,H}\left(\delta(1-\varepsilon)\right)\leqslant -\log\mathbb{P}\left(N(H,\mathbb{G}(n,p))\geqslant(1+\delta)n^{v_H}p^{e_H}\right)\leqslant(1+\varepsilon)\Psi_{p,H}\left(\delta(1+\varepsilon)\right).$$

The theorem immediately preceding this conjecture establishes the analogous mean-field approximation for stars $H=K_{1,r}$ in a different sparsity regime. The conjecture proposes that the same approximation holds for every strictly balanced graph in the stated regime, extending the known result for irregular subgraph counts and identifying the variational problem as the correct logarithmic upper-tail scale.