Random Matrices and Related Topics in Jeju

How Jack polynomials and corresponding generalised hypergeometric functions enter random matrix through the consideration of parameter dependent Gaussian Hermitian random matrices will be revised. As an application, a class of probability density functions relating to particular joint moments for the circular beta ensemble will be evaluated. This allows the special value $(e^2-5)/(4\pi)$ known from a number theory problem to be reclaimed.

A basic result of random matrix theory is the circular law: the eigenvalues of complex matrices with i.i.d. entries fill out a disk as the size of the matrices increases. In the case of the Ginibre ensemble with point insertions there is a repulsion of eigenvalues away from the inserted points. In the large size limit fill out a region known as the droplet, that is different from a disk. The zeros of the average characteristic polynomial are believed to converge to a certain contour inside the droplet, and their limiting zero counting measure is called the mother body.
We study the polynomials and the mother body for the case of two insertions that are symmetrically located on the imaginary axis and for a small time parameter. Then we characterize the mother body by means of a vector equilibrium problem for two measures. We prove strong asymptotics of the polynomials by means of a steepest descent analysis of a matrix valued Riemann-Hilbert problem of size $3\times3.$ The RH problem comes from the fact that the polynomials not only satisfy planar orthogonality, but are also multiple orthogonal of type I on a contour in the complex plane.
The talk is based on joint work with Mario Kieburg and Sampad Lahiry.

The ground state of $N$ noninteracting Fermions in a rotating harmonic trap enjoys a one-to-one mapping to the complex Ginibre ensemble. This setup is equivalent to electrons in an electric field described by Landau levels. The mean and variance of the number of particles in a circular domain can be computed exactly for finite $N$ and in three different large-$N$ limits. In the bulk and at the edge of the spectrum the result is universal for a large class of rotationally invariant potentials. The same can be proven for the quaternionic Ginibre ensemble and its corresponding generalisation. For the real Ginibre ensemble we determine the large-$N$ limit at the origin and conjecture its universality in the bulk and at the edge.

The singular values as well as the usually complex eigenvalues of a generic fixed complex square matrix are the most natural spectral quantities to be studied. Hence, their interrelation has fascinated mathematicians for more than a century. Unfortunately, apart from a single equality there are generally only inequalities which link the two objects. This drastically changes when studying random matrices whose singular vectors are Haar distributed, as we have shown and provided an explicit formula for their joint probability densities a few years ago. Evidently, eigenvalues and singular values will be correlated. The question is how much and in what way. For this reason, we derived and studied the correlation function between one eigenvalue and one eigenvector. In my talk, I will report on these results and what insights they are giving us. This talk is based on a work with Matthias Allard.

In this talk, we will introduce the random band matrix conjecture and some recent work on this topic. The conjecture predicts a phase transition of the eigenvector behaviors in the high-dimensional case. Parallel to this, the Anderson conjecture unveils similar phenomena within disordered systems. Our study used some non-field random matrix models to provide deeper insights into these complex phenomena. It is a joint work with Fan Yang.

Hermite and Laguerre $\beta$ ensembles are two of the most well studied models in random matrix theory with the special cases $\beta=1,2,4$ corresponding to eigenvalue distributions of classical Gussian and Wishart ensembles. Using tridiagonal matrix models for these ensembles Ramirez, Rider and Virag showed that the scaling limit of the largest eigenvalue is the Tracy-Widom $\beta$ distribution whose upper tail decays like $\exp(-2\beta t^{3/2}/3)$ while the lower tail decays like $\exp(-\beta t^{3}/24)$. I shall describe some recent work establishing sharp tail estimates in the pre-limiting models up to $(1+o(1))$ error at the level of exponents, improving earlier results by Ledoux and Rider. I shall also describe a number of applications of these estimates to exactly solvable planar last passage percolation, which are related to the classical $\beta$ ensembles via some remarkable distributional identities.

In the study of extreme eigenvalues of Wigner matrices and the largest eigenvalue of sample covariance matrices, it has been established that a weak 4th moment condition is necessary and sufficient for the Tracy-Widom law to hold. In this talk, we will show that the Tracy-Widom law is more robust for the smallest non-zero eigenvalue of the sample covariance matrix. We will specifically illustrate a phase transition from the Tracy-Widom distribution to a Gaussian distribution when the tail exponent of the matrix entry distribution crosses the value of 8/3. This talk is based on a joint work with Jaehun Lee and Xiaocong Xu.

Let $A$ be the adjacency matrix of the Erdős-Rényi directed graph $G(N,p)$. For $N^{−1+o(1)}\leq p\leq 1/2$, we prove that near the unit circle, the local eigenvalue statistics of $A/\sqrt{Np(1−p)}$ coincide with those of the real Ginibre ensemble. As a by-product, we also show that all non-trivial eigenvectors of $A$ are completely delocalized. For Hermitian random matrices, it is known that the edge statistics are sensitive to sparsity: in the very sparse regime, one needs to remove many noise random variables (which affect both the mean and the fluctuation) to recover the Tracy-Widom distribution. Our results imply that, compared to their analogues in the Hermitian case, the edge statistics of non-Hermitian sparse random matrices are more robust.

The normal matrix model or two-dimensional one-component plasma at a specific temperature is a particle system in the plane with logarithmic interactions. In equilibrium the particles form distinct droplets when placed in an external potential. Using the Riemann-Hilbert approach we determine, in a specific external potential, the local statistical behaviour of the particles at the point where two droplets merge and observe an anisotropic scaling behaviour with particles being much further apart in the direction of merging than the perpendicular direction. At the merging singularity the local statistics is related to the solution of the Painlevé II equation. In its vicinity Ginibre bulk and edge statistics, as well as the sine-kernel and universality class corresponding to the elliptic ensemble in the weak non-Hermiticity regime are observed. This is joint work with Meng Yang and Seung-Yeop Lee.

Consider $D$ random systems that are modeled by independent $N\times N$ complex Hermitian Wigner matrices. Suppose they are lying on a circle and the neighboring systems interact with each other through a deterministic matrix $A$. We prove that in the asymptotic limit $N\to \infty$, the whole system exhibits a quantum chaos transition when the interaction strength $\|A\|_{HS}$ varies. Specifically, when $\|A\|_{HS}\ge N^{\epsilon}$, we prove that the bulk eigenvalue statistics match those of a $DN\times DN$ GUE asymptotically and each bulk eigenvector is approximately equally distributed among the $D$ subsystems with probability $1-o(1)$. These phenomena indicate quantum chaos of the whole system. In contrast, when $\|A\|_{HS}\le N^{-\epsilon}$, we show that the system is integrable: the bulk eigenvalue statistics behave like $D$ independent copies of GUE statistics asymptotically and each bulk eigenvector is localized on only one subsystem. Based on joint work with Bertrand Stone and Jun Yin.

Recently, motivated by problems in different fields such as numerical analysis, semiclassical analysis, and mathematical physics, there has been an increasing interest in understanding the spectral properties of random perturbations of non-normal operators. In this talk we will consider a random perturbation of a non-normal Toeplitz matrix, and discuss the localization and delocalization properties of its eigenvectors, which is determined by the strength of the perturbation. Based on joint work with Martin Vogel and Ofer Zeitouni.

What is the large deviation behaviour of the spectrum of a Rademacher Wigner matrix? Owing to the non-integrability of the model, this question has been challenging the classical large deviation theory for decades. Motivated by this question, we examine a sparse relaxation of this problem, by zeroing out independently each entry on the upper triangular part with a rate going to 0 with the dimension, in such a way that the average number of non-zero entries on each line diverges at least as the logarithm of the dimension. In this sparsity regime, we obtain a large deviations principle for the empirical spectral measure of diluted Wigner matrices with bounded entries with a rate function solution of a certain variational problem. The rate function reveals in particular that the only possible deviations are around measures coming from Quadratic Vector Equation. As a byproduct, we obtain a large deviations principle for the empirical spectral measure of sparse supercritical Erdos-Renyi graphs.

The periodic KPZ fixed point is the conjectural universal limit of the KPZ universality class models on a ring when the period and time both tend to infinity in a certain critical way. For the case of the periodic narrow wedge initial condition, we consider the conditional distribution when the periodic KPZ fixed point is unusually large at a particular position and time. We prove a conditional limit theorem up to the “pinchup” time. For the KPZ fixed point itself, this limit was computed by Liu and Wang in 2022. We identify the regimes where the limit changes from the KPZ fixed point and find probabilistic descriptions. This is a joint work with Zhipeng Liu.

I will discuss results for the derivative of the characteristic polynomial of a unitary matrix drawn at random from the Circular Unitary Ensemble (CUE). We obtain a formula for the non-integer moments in terms of a confluent hypergeometric function, valid in the limit of large matrix dimension. The approach is based on the theory of log-correlated Gaussian fields. In the integer moment case our formula reduces to a single Laguerre polynomial and we are able to describe the structure of asymptotic corrections. I will discuss possible implications of our results for derivative moments of the Riemann zeta function. This is joint work with Fei Wei (University of Oxford).

We investigate a very general technique to obtain CLTs with near-optimal rates of convergence for broad classes of strongly dependent stochastic systems, based on the zeros of the characteristic function. Using this, we demonstrate Gaussian fluctuations for the magnetization (i.e., the total spin) for a large class of ferromagnetic spin systems on Euclidean lattices, in particular those with continuous spins, at the near-optimal rate of $O(\log |\Lambda| · |\Lambda|−1/2)$ for system size |Λ|. This includes, in particular, the celebrated $XY$ and Heisenberg models under ferromagnetic conditions. Our approach leverages the classical Lee-Yang theory for the zeros of partition functions, and subsumes as a special case a technique of Lebowitz, Ruelle, Pittel and Speer for deriving CLTs in discrete statistical mechanical models, for which we obtain sharper convergence rates. In a very different application, we obtain CLTs for linear statistics of a wide class of point processes known as $\alpha$-determinantal processes which interpolate between negatively and positively associated random point fields (including the usual determinantal, permanental and Poisson processes). Notably, we address strongly correlated processes in dimensions $\ge 3$, where connections to random matrix theory are not available, and handle a broad class of kernels including those with slow spatial decay (such as the Bessel kernel in general dimensions). A key ingredient of our approach is a broad, quantitative extension of the classical Marcinkiewicz Theorem that we establish under the significantly milder condition that the characteristic function is non-vanishing only on a bounded disk. Joint work with T.C. Dinh, H.S. Tran and M.H. Tran.

The Fyodorov-Hiary-Keating conjecture has two parts, one in random matrix theory and one about the Riemann zeta function. In the random matrix part, it gives the precise distributional limit for the maximum of a characteristic polynomial of a Haar Unitary matrix. Using the replica method and a physically motivated freezing ansatz, they derived one of the most precise log-correlated field predictions to date, and they did it for a process which was not even Gaussian. While existing work shows that Haar Unitary matrices had many log correlated field connections, techniques for showing convergence of the maximum typically rely on either the Gaussianity of the underlying process or precise branching structures built into the problem; the characteristic polynomial has neither. We will describe the problem and the current state of the art, in which we (the speaker and Ofer Zeitouni) show the convergence in law of the maximum of a Circular-beta ensemble random matrix to a convolution of a Gumbel and the total mass of a (non-Gaussian) critical multiplicative chaos.

We show how to obtain an asymptotic expansion of orthogonal polynomials in the planar setting with exponentially varying weights.

I discuss the asymptotic behavior of the determinants of Toeplitz and Toeplitz plus Hankel matrices, and of the finite sections of functions of Toeplitz operators, and show how easily operator theory can be used to produce a number of results about these classes in many situations. The focus is on matrix-valued symbols and includes both smooth and singular symbols. Some applications will also be mentioned. Joint work with Estelle Basor and Torsten Ehrhardt.

The Pearcey kernel is a classical and universal kernel arising from random matrix theory, which describes the local statistics of eigenvalues when the limiting mean eigenvalue density exhibits a cusp-like singularity. It appears in a variety of statistical physics models beyond matrix models as well. We consider the Fredholm determinant of a trace class operator acting on $L^2 (-s, s)$ with the Pearcey kernel. We establish an integral representation for the determinant, which involves the Hamiltonian associated with a family of special solutions to a system of nonlinear differential equations. We also derive large gap asymptotics for the determinant and obtain asymptotic statistical properties for the related point processes. This is a joint work with Shuai-Xia Xu and Lun Zhang.

Consideration in this talk is a universal phenomenon arising from the random matrix theory, namely, the hard-to-soft edge transition. By showing that the symmetrically transformed Bessel kernel admits a full asymptotic expansion for the large parameter, we establish a hard-to-soft edge transition expansion. The talk is based on a joint work with Luming Yao.

This talk concerns orthogonal polynomials with respect to area measure on domains in the complex plane, also known as Bergman polynomials. The study of their large-degree asymptotics is a classical topic going back to work of Carleman in the 1920's, and they have recently received increased interest from the random matrix community due to connections to 2D Coulomb gas models. I will discuss Bergman polynomials for domains whose boundary is a piecewise analytic Jordan curve without cusps. In this case, the zeros of the associated Bergman polynomials exhibit a curious dichotomy, determined by the analytic continuation properties of the interior conformal maps of the domain. The talk is based on recent joint work with Erwin Miña-Díaz (arXiv:2404.09335).

I will discuss large deviation principles for the right-most eigenvalue of Wigner matrices with sub-Gaussian entries. Previous work of Guionnet and Husson established a universal rate function for the light-tailed "sharp sub-Gaussian" case, where large deviations result from "delocalized" tilting strategies. On the other hand, large deviations for heavy-tailed matrices and sparse graphs and networks have been shown to be governed by "localization" phenomena. The general sub-Gaussian case is in some sense critical, with non-universal rate functions determined by a mixture of competing localized and delocalized tilts. Our approach also leads to information on the conditional structure of the associated eigenvector. Based on joint work with Raphael Ducatez and Alice Guionnet.

Matrix perturbation theory is a foundational subject across multiple disciplines, including probability, statistics, machine learning, and applied mathematics. Perturbation bounds play a critical role in quantifying the impact of small noise on matrix spectral parameters in various applications such as matrix completion, principal component analysis, and community detection. In this talk, we focus on the additive perturbation model, where a low-rank data matrix is perturbed by Gaussian noise. We provide a comprehensive analysis of the singular subspaces, extending the classical Davis-Kahan-Wedin theorem and offering fine-grained analysis of the singular vector matrices. Also, we will show the practical implications of our perturbation results, specifically highlighting their application in Gaussian mixture models.

We consider 2D Coulomb gases with the external potential $Q(z)=|z|^2-2c \log|z-a|$, where $c>0$ and $a \in \mathbb{C}$. Equivalently, this model can be realised as $N$ eigenvalues of the complex Ginibre matrix of size $(c+1) N \times (c+1) N$ conditioned to have deterministic eigenvalue $a$ with multiplicity $cN$. Depending on the values of $c$ and $a$, the droplet reveals a phase transition: it is doubly connected in the post-critical regime and simply connected in the pre-critical regime. In both regimes, we derive precise large-$N$ expansions of the free energy up to the $O(1)$ term, providing a non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture made for general planar Coulomb gas ensembles. As a consequence, our results provide asymptotic behaviours of moments of the characteristic polynomial of the complex Ginibre matrix, where the powers are of order $O(N)$. Furthermore, by combining with a duality formula, we obtain precise large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. This talk is based on the joint work with Sung-Soo Byun and Seong-Mi Seo.

We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given planar stationary point process. We characterize those stationary processes with finite second moment for which, after subtracting the mean, the random function becomes stationary. These random meromorphic functions can be viewed as random analogues of the Weierstrass zeta function from the theory of elliptic functions, or equivalently as electric fields generated by an infinite random distribution of point charges. The talk will be based on joint work with Oren Yakir and Aron Wennman (arXiv, 2022).

We consider two-dimensional Coulomb systems in the regime when the droplet is connected, while the coincidence set for the obstacle problem contains an analytic Jordan curve outside of the droplet. A nontrivial (Heine-distributed) number of particles will tend to fall in the vicinity of this curve, which we denote "spectral outpost". Under the process of Laplacian growth, the outpost grows into a new ring-shaped component of the droplet, and the study of outposts is therefore closely related to the regime of disconnected droplets. We study among other things fluctuations of linear statistics, which interestingly are non-Gaussian. The talk is based on joint works with Joakim Cronvall and Christophe Charlier.

Motivated by the recent study of the non-Hermitian matrix-valued Brownian motion (Burda et al.: Nucl. Phys. B 897, 421 (2015)), we propose two types of matrix-valued dynamical processes generated by nonnormal Toeplitz matrices with additive perturbations. We first show the complicated structures and motions of "eigenvalues" found in numerical calculations. Then we derive the specific equations which determine the exact-eigenvalue processes. Comparing the solutions of these equations with the numerical results, coexistence of exact eigenvalues and pseudospectra is clarified. We characterize the numerical results using the notion of symbol curves of the corresponding nonnormal Toeplitz operators represented by infinite matrices. We report a new phenomenon in our second model such that at each time the outermost closed simple curve cut out from the symbol curve is realized as exact eigenvalues, but the inner part of symbol curve is reduced in size and embedded as a complicated structure in the pseudospectrum. The asymptotics of the processes in the infinite-matrix limit are also discussed. This talk is based on the joint work with Saori Morimoto (Chuo) and Tomoyuki Shirai (Kyushu). A preprint is available at https://arxiv.org/abs/2401.08129.

The zeros of random power series with i.i.d. complex Gaussian coefficients form the determinantal point process associated with the Bergman kernel. As a natural generalization of this model, we are concerned with Gaussian power series with coefficients being stationary, centered, complex Gaussian process. We focus on the zeros of such analytic Gaussian power series and discuss the asymptotic behavior of their density as the observation window approaching to the convergence circle.

We study the Brownian evolution of large non-Hermitian matrices and show that their log-determinant converges to a 2+1 dimensional Gaussian field in the Edwards-Wilkinson regularity class, i.e. logarithmically correlated for the parabolic distance. This gives a dynamical extension of the celebrated result by Rider and Virag (2007) proving that the fluctuations of the eigenvalues of Gaussian non-Hermitian matrices converge to a 2 dimensional log-correlated field. Our result, previously not known even in the Gaussian case, holds out of equilibrium for general matrices with i.i.d. entries as an initial condition. We also study the extremal values of these fields and demonstrate their logarithmic dependence on the matrix dimension.

In this talk, I will describe work on the theory of generalized quadrature domains (GQDs), specifically those with respect to weighted area measures and those with Abelian quadrature functions. We develop a method of approaching uniqueness problems for GQDs via their correspondence with Hele-Shaw flow, and use of the Faber transform. Applications to the integrability of the Hele-Shaw flow are also discussed. This is joint work with Nikolai Makarov.

We study both smooth and rough linear statistics of eigenvalues of the complex elliptic Ginibre ensemble depending on a parameter $0\leq\tau$ < 1. In strong non-Hermiticity ($\tau$ fixed) it is known (for general random normal matrix models) by Ameur, Hedenmalm and Makarov that the smooth linear statistics satisfy a CLT in the large $n$ limit, and we present an explicit formula for the limiting variance. Furthermore, we show that in the case of rough linear statistics the number variance is proportional to the perimeter of the set under consideration. In the weak non-Hermiticity regime $\tau=1-\kappa/n^a$, with choice $a=1$, we show that the number variance exhibits features of the number variance of both non-Hermitian matrices as well as Hermitian matrices (as in Costin-Lebowitz). Lastly, when 0 < $a$ < 1, we show that the limiting variance of (rescaled) smooth linear statistics interpolates between that of the Ginibre ensemble and the GUE. This is joint work with Gernot Akemann and Maurice Duits.

In this talk, I will discuss large $N$ expansions for the partition functions of 2D determinantal and Pfaffian Coulomb gases with radially symmetric external potentials. We find that some correction terms in the expansion represent the geometric and topological properties of the region where particles accumulate. This talk is based on joint work with Sung-Soo Byun and Nam-Gyu Kang.