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 ($GM{C}^{\gamma }$) 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 $\gamma =\sqrt{2/\beta }$, which we establish when $\gamma \le 1$ or equivalently $\beta \ge 2$. This corresponds to the subcritical and critical phases for $GM{C}^{\gamma }$. As a side product, we answer positively a question raised by Virág, on the fractal spectrum of a random measure constructed from $C\beta E$. We also give an alternative proof of the Fyodorov–Bouchaud formula concerning the total mass of $GM{C}^{\gamma }$ on the circle. This conjecture was recently settled by Remy using Liouville conformal field theory. We can go even further and give an explicit description of the Fourier coefficients of the random measure $GM{C}^{\gamma }$ in terms of independent Beta random variables. 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 $GM{C}^{\gamma }$ corresponds to $\beta =2$, where the geometry and representation theory of unitary groups lie.