The universal maximum conjecture for log-correlated fields

Let $X(n)=\{X_v(n),v\in V_n\}$ be a log-correlated field in the sense of logarithmic covariance, with $E[X_v(n)]=0$ and $E[X_v(n)^2]\approx\sigma^2n$. Set

$$c=\sqrt{2\log 2}\,\sigma,\qquad a_n=cn-\frac{3}{2}\frac{\sigma^2}{c}\log n,\qquad b_n=1.$$

Universal maximum conjecture. The maximum satisfies, for some constant $C$ and a random variable $Z$ called the derivative martingale,

$$\lim_{n\to\infty}P\left(\max_{v\in V_n}X_v(n)\leq a_n+b_nx\right)=E\left[e^{-CZ e^{-cx}}\right].$$

This prediction expresses the expected universal extreme-value law for log-correlated fields that are sufficiently close to Gaussian fields. It is motivated by branching-random-walk heuristics and is intended to apply across a broad class of models, although the required hypotheses and the general proof remain open.