Mean-field approximation for irregular strictly balanced subgraph upper tails

Establish that, for every strictly balanced graph H with maximum degree Δ, every ε∈(0,1), every δ>0, and edge probabilities p satisfying p≪n^{-1/Δ} and n^{v_H}p^{e_H}≫(log n)^{α_H^*/(α_H^*−1)}, the logarithm of the upper-tail probability for N(H,G(n,p)) is asymptotically bounded above and below by the naïve mean-field variational quantities Ψ_p(N(H,·),δ(1−ε)) and Ψ_p(N(H,·),δ(1+ε)), respectively, up to multiplicative factors 1−ε and 1+ε.

Background

The paper proves that the naïve mean-field variational approximation is asymptotically tight for upper tails of irregular subgraph counts in the regime n{-1/Δ}≪p≪1. It also proves the same type of approximation for the r-armed star K_{1,r} in the distinct intermediate regime p≪n{-1/r} with n{r+1}pr≫(log n){r/(r−1)}. The conjecture asks whether this mean-field behavior extends from stars to every strictly balanced graph in that intermediate regime.

Here, Ψ_p(N(H,·),·) is the variational cost obtained by minimizing the relative-entropy cost of an inhomogeneous Erdős–Rényi edge-probability vector subject to the expected number of copies of H exceeding the specified upper-tail threshold. The conjecture leaves the result unresolved for general strictly balanced graphs when p≪n{-1/Δ} and n{v_H}p{e_H} is above the Poisson-regime threshold.

References

In light of Theorem \ref{thm:mean-field} we make the following plausible conjecture. \begin{conjecture} For any strictly balanced graph $H$, $\varepsilon \in (0,1)$, $\delta >0$, and $p$ such that $p \ll n{-1/\Delta}$ and $n{v_{H} p{e_{H} \gg (\log n){\alpha_H/(\alpha_H^-1)}$, and for all large $n$

(1-\varepsilon)\Psi_{p}\left(N(H), \cdot),\delta(1-\varepsilon)\right) -\logP\left(N(H, G(n,p))(1+\delta)n{r+1}p{r}\right) (1+\varepsilon)\Psi_{p}\left(N(H), \cdot),\delta(1+\varepsilon)\right).

\end{conjecture}

— Upper tail bounds for irregular graphs  (2503.05311 - Basak et al., 7 Mar 2025) in Conjecture following Theorem 1.4, subsection “Naïve mean-field approximation” (Section 1)