On the circle, Gaussian Multiplicative Chaos and Beta Ensembles match exactly
Reda Chhaibi, Joseph Najnudel
TL;DR
The work proves an exact equivalence between Gaussian Multiplicative Chaos on the circle and the limiting Circular Beta Ensemble measure for β ≥ 2, via a spectral approach rooted in orthogonal polynomials on the unit circle. By coupling Verblunsky coefficients with Beta-distributed moduli and carefully renormalizing, GMC^γ with γ = sqrt(2/β) matches the limiting CβE measure μ^β, and the total mass distributions, Fourier coefficients, and Hausdorff dimensions are explicitly described. The authors develop universal trace bounds, convergence of OPUC to a log-correlated Gaussian field, and a diffusive limit for the modulus dynamics, culminating in a stochastic-differential-equation framework that yields both existence and uniqueness of the limiting law. Their results connect GMC with random matrix theory and conformal field theory perspectives, providing explicit moment formulas and suggesting deep links to KPZ, KPZ duality, and Liouville CFT integrability. These insights yield a robust spectral regularization of GMC by random matrix ensembles and open avenues for extending the correspondence beyond subcritical regimes and toward broader fractal and CFT-related structures.
Abstract
We identify an equality between two objects arising from different contexts of mathematical physics: Kahane's Gaussian Multiplicative Chaos ($GMC^γ$) on the circle, and the Circular Beta Ensemble $(CβE)$ 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 $γ= \sqrt{\frac{2}β}$, which we establish only when $γ\leq 1$ or equivalently $β\geq 2$. 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 $GMC^γ$ on the circle. This conjecture was recently settled by Rémy 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 $β>0$, thanks to existing spectral theory. Remarkably, the critical parameter for $GMC^γ$ corresponds to $β=2$, where the geometry and representation theory of unitary groups lie.