Fyodorov–Keating conjecture on thick points of CUE characteristic polynomials

Let $U_N$ be a random matrix distributed according to the Haar measure on the group of $N\times N$ unitary matrices, and let $p_N(\theta)=\det(1-U_N e^{-i\theta})$ be its characteristic polynomial restricted to the unit circle $\{e^{i\theta}:\theta\in\mathbb{T}\}$. Let $G$ denote the Barnes $G$-function. Fyodorov–Keating conjecture. As $N\to\infty$, for any $0<\gamma<1$, the random variable

$$\frac{\frac{1}{2\pi}\int_0^{2\pi}\mathbf{1}\{\log |p_N(\theta)|>\gamma\log N\}\,d\theta}{N^{-\gamma^2}\frac{1}{\sqrt{\pi\log N}}\frac{G(1+\gamma)^2}{2\gamma G(1+2\gamma)}\frac{1}{\Gamma(1-\gamma^2)}}$$

converges in distribution to a positive random variable with density

$$\mathcal{P}_\gamma(x)=\gamma^{-2}x^{-1-\gamma^{-2}}e^{-x^{-\gamma^{-2}}}\mathbf{1}\{x>0\}.$$

This conjecture exemplifies the expected universal connection between multiplicative chaos measures and fluctuations of thick points for logarithmically correlated fields. Results are known for several specific models, but identifying the law of this random variable in terms of Gaussian multiplicative chaos remains an open problem in the general setting.