For an N × N Haar distributed random unitary matrix UN , we consider the random field defined by counting the number of eigenvalues of UN in a mesoscopic arc centered at the point u on the unit circle. We prove that after regularizing at a small scale ǫN > 0, the renormalized exponential of this field converges as N → ∞ to a Gaussian multiplicative chaos measure in the whole subcritical phase. We discuss implications of this result for obtaining a lower bound on the maximum of the field. We also show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in [55]. By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. Our approach to the L 1 -phase is based on a generalization of the construction in Berestycki [5] to random fields which are only asymptotically Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context.

We consider 2-dimensional determinantal processes that are rotationinvariant and study the fluctuations of the number of points in disks. Based on the theory of mod-phi convergence, we obtain Berry–Esseen as well as precise moderate to large deviation estimates for these statistics. These results are consistent with the Coulomb gas heuristic from the physics literature. We also obtain functional limit theorems for the stochastic process $(\# D_r)_{r>0}$ when the radius $r$ of the disk $D_r$ is growing in different regimes. We present several applications to invariant determinantal processes, including the polyanalytic Ginibre ensembles, zeros of the hyperbolic Gaussian analytic function, and other hyperbolic models. As a corollary, we compute the precise asymptotics for the entanglement entropy of (integer) Laughlin states for all Landau levels.

The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices -the Gaussian Unitary Ensemble being the prime example of such an ensemble.Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher-Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.

We study mesoscopic linear statistics for a class of determinantal point processes which interpolates between Poisson and Gaussian Unitary Ensemble (GUE) statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the GUE eigenvalue process. An example of such a system comes from considering the distribution of non-colliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the GUE kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we get a central limit theorem (CLT) of either Poisson or GUE type. On the other hand, in the critical regime where the scales are the same, we get a non-Gaussian process in the limit. Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics. These results rely on an asymptotic sine-kernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process. Analogous determinantal processes on the circle are also considered with similar results.

We study the fluctuations of certain biorthogonal ensembles for which the underlying family {P, Q} satisfies a finite-term recurrence relation of the form xP (x) = JP (x). For polynomial linear statistics of such ensembles, we reformulate the cumulants' method introduced in [Sos00a] in terms of counting lattice paths on the graph of the adjacency matrix J. In the spirit of [BD], we show that the asymptotic fluctuations of polynomial linear statistics are described by the right-limits of the matrix J. Moreover, whenever the right-limit is a Laurent matrix, we prove that the CLT obtained in [BD] is equivalent to Soshnikov's main combinatorial lemma. We discuss several applications to unitary invariant Hermitian random matrices. In particular, we provide a general Central Limit Theorem (CLT) in the one-cut regime. We also prove a CLT for square singular values of product of independent complex rectangular Ginibre matrices. Finally, we discuss the connection with the Strong Szegő theorem where this combinatorial method originates.