On the circle, Gaussian Multiplicative Chaos and Beta Ensembles match exactly

Abstract

We identify an equality between two objects arising from different contexts of mathematical physics: Kahane's Gaussian Multiplicative Chaos ($GMC^\gamma$) on the circle, and the Circular Beta Ensemble $(C\beta E)$ from Random Matrix Theory. This is obtained via an analysis of related random orthogonal polynomials, making the approach spectral in nature. In order for the equality to hold, the simple relationship between coupling constants is $\gamma = \sqrt{\frac{2}{\beta}}$, which we establish only when $\gamma \leq 1$ or equivalently $\beta \geq 2$. This corresponds to the sub-critical and critical phases of the $GMC$. As a side product, we answer positively a question raised by Virag. We also give an alternative proof of the Fyodorov-Bouchaud formula concerning the total mass of the $GMC^\gamma$ on the circle. This conjecture was recently settled by R\'emy using Liouville conformal field theory. We can go even further and describe the law of all moments. Furthermore, we notice that the ``spectral construction'' has a few advantages. For example, the Hausdorff dimension of the support is efficiently described for all $\beta>0$, thanks to existing spectral theory. Remarkably, the critical parameter for $GMC^\gamma$ corresponds to $\beta=2$, where the geometry and representation theory of unitary groups lie.