Fyodorov–Keating supercritical moment asymptotic conjecture for C$\beta$E

Let $\beta>0$, $k\in\mathbb{N}$, and $s\in\mathbb{R}_+$. Let $\mathsf{M}^{(\beta)}_N(k;s)$ denote the corresponding C$\beta$E field moment, and let $\mathfrak{c}^{(\beta)}(k;s)$ be the leading-order constant. In the supercritical regime $2ks^2>\beta$, Fyodorov–Keating supercritical moment conjecture. As $N\to\infty$,

$$\mathsf{M}^{(\beta)}_N(k;s)\sim \mathfrak{c}^{(\beta)}(k;s)N^{2k^2s^2\beta^{-1}-k+1}.$$

This prediction concerns moments beyond the range governed by Gaussian multiplicative chaos. The source attributes it to Fyodorov, Keating, and related work, and notes that the special case $\beta=2$ appears as a stated prediction; unlike the moment-critical case, no general proof is reported here.