The goal of this article is to expand on the relationship between random matrix and multiplicative chaos theories using the integrability properties of the circular β-ensembles. We obtain the multiplicative chaos convergence for the characteristic polynomial and eigenvalue counting function of the circular β-ensembles throughout the subcritical phase, including negative powers. This generalizes recent results in the unitary case, [8, 40], to any β>0 and for the eigenvalue counting field.

We consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on (Formula presented.), with a smooth confining potential. In the semiclassical limit, where the number of particles tends to infinity, we obtain a Szegő-type central limit theorem for the fluctuations of smooth linear statistics. More precisely, the Laplace transform of any statistic converges without renormalisation to a Gaussian limit with a (Formula presented.) -type variance, which depends on the potential. In the one-well (one-cut) case, using the quantum action-angle theorem and additional micro-local tools, we reduce the problem to the asymptotics of Fredholm determinants of certain approximately Toeplitz operators. In the multi-cut case, we show that for generic potentials, a similar result holds and the contributions of the different wells are independent in the limit.

We study local asymptotics for the spectral projector associated to a Schrödinger operator ħ² ∆ + V on Rⁿ in the semiclassical limit as ħ → 0. We prove local uniform convergence of the rescaled integral kernel of this projector towards a universal model, inside the classically allowed region as well as on its boundary. This implies universality of microscopic fluctuations for the corresponding free fermions (determinantal) point processes, both in the bulk and around regular boundary points. Our results apply to a general class of smooth potentials in arbitrary dimension n ≥ 1. These results are complemented by studying both macroscopic and mesoscopic fluctuations of the point process. We obtain tail bounds for macroscopic linear statistics and, provided n ≥ 2, a central limit theorem for both macroscopic and mesoscopic linear statistics in the bulk.

For any integer m < n, where m can depend on n, we study the rate of convergence (Formula Presented) to its limiting Gaussian as n → ∞ for orthogonal, unitary and symplectic Haar distributed random matrices U of size n. In the unitary case, we prove that the total variation distance is less than (Formula Presented) times a constant. This result interpolates between the super-exponential bound obtained for fixed m and the 1/n bound coming from the Berry–Esseen theorem applicable when m ≥ n by a result of Rains. We obtain analogous results for the orthogonal and symplectic groups. In these cases, our total variation upper bound takes the form (Formula Presented) times a constant and the result holds provided n > 2m. For m = 1, we obtain complementary lower bounds and precise asymptotics for the L2-distances as n → ∞, which show how sharp our results are.

We prove that multiplicative chaos measures can be constructed from extreme level sets or thick points of the underlying logarithmically correlated field. We develop a method which covers the whole subcritical phase and only requires asymptotics of suitable exponential moments for the field. As an application, we establish that these estimates hold for the logarithm of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix (CUE), using known asymptotics for Toeplitz determinant with (merging) Fisher–Hartwig singularities. Hence, this proves a conjecture of Fyodorov and Keating concerning the fluctuations of the volume of thick points of the CUE characteristic polynomial.

For any integer $m<n$, where $m$ can depend on $n$, we study the rate of convergence of $\frac{1}{\sqrt{m}}\tr \mathbf{U}^m$ to its limiting Gaussian as $n\to\infty$ for orthogonal, unitary and symplectic Haar distributed random matrices $\mathbf{U}$ of size $n$. In the unitary case, we prove that the total variation distance is less than $\Gamma(\floor{n/m}+2)^{-1}m^{-\floor{n/m}}\floor{n/m}^{1/4}\sqrt{\log n}$ times a constant. This result interpolates between the super-exponential bound obtained for fixed $m$ and the $1/n$ bound coming from the Berry-Esseen theorem applicable when $m\ge n$ by a result of Rains. We obtain analogous results for the orthogonal and symplectic groups. In these cases, our total variation upper bound takes the form $\Gamma(2\floor{n/m}+1)^{-1/2}m^{-\floor{n/m}+1}(\log n)^{1/4}$ times a constant and the result holds provided $n \geq 2m$. For $m=1$, we obtain complementary lower bounds and precise asymptotics for the $L^2$-distances as $n\to\infty$, which show how sharp our results are.