Fyodorov–Keating conjecture on moments of regularized counting statistics

Let $Z_N(\theta)$ be the regularized counting-statistics field and let $\gamma>0$. Define

$$M_N^{\gamma}:=\int_0^{2\pi}e^{\gamma Z_N(\theta)}\,d\theta.$$

For any $q\in\mathbb{N}$ such that $\gamma^2q<2$, let $C_{\mu,q}$ denote the normalization constant appearing in the conjectured moment asymptotics, let $G(z)$ be the Barnes $G$-function, and set

$$C_{\gamma,q}=\frac{G(1+\gamma/\sqrt{2})^{2q}}{G(1+\gamma\sqrt{2})^q}.$$

Fyodorov–Keating conjecture.

$$\lim_{N\to\infty}N^{-\frac{\gamma^2q}{2}}\mathbb{E}[(M_N^{\gamma})^q] =C_{\mu,q}\int_{[0,2\pi]^q}\prod_{1\leq j<k\leq q}|e^{i\theta_j}-e^{i\theta_k}|^{-\gamma^2}\,d\theta_1\cdots d\theta_q =C_{\gamma,q}(2\pi)^q\frac{\Gamma(1-\gamma^2q/2)}{\Gamma(1-\gamma^2/2)^q}.$$

This conjecture concerns the total mass of the exponential of the log-correlated field arising from regularized random-matrix counting statistics and is related to predictions for the extreme values of that field. The source states that, for $q\neq1,2$, the conjecture remains an open problem; the row is therefore open.