2023 Seminars

Abstract: The elliptic Ginibre ensemble belongs to a random matrix ensemble which interpolates between the Ginibre and Gaussian orthogonal ensembles. A common characteristic of the ensembles is a presence of the real eigenvalues. The real eigenvalue statistic of the GinOE is distinct from that of the GOE, and it gives rise to two different regimes according to the non-Hermiticity parameter. In this talk, we will present the finite size corrections of the real eigenvalue densities for the elliptic Ginibre ensembles. We discuss finite size corrections for the limiting densities for the global and the edge scaling limits in the both regimes. Especially, the rates of convergence to the universal limit are shown to be dependent on the non-Hermiticity parameter. This is based on a joint work with Sung-Soo Byun.

Abstract: Let X be an N x M signal matrix with i.i.d. real entries drawn from some distribution. We consider a statistical model for the measurement Y of XX* through an additive Gaussian channel in the high-dimensional regime where M scales with N as M=o(N^{1/4}). Working in the Bayes-optimal setting, we show that the limiting free entropy of the model, equivalently the mutual information between the measurement Y and signal X, is given by a variational formula involving a replica symmetric potential corresponding to an M-dimensional vector channel. In fact, we show that in many cases, we can reduce further to the replica symmetric potential of a scalar channel (M = 1). Our arguments draw on an application of the cavity method allowing for growing rank M, a surprisingly simple result on overlap concentration, and some information-theoretic identities concerning concavity properties of the distribution on the entries of X.

Abstract: Random matrix theory enjoys an intimate connection with various branches of mathematics. One prominent illustration of this relationship is the Harer-Zagier formula, which serves as a well-known example demonstrating the combinatorial and topological significance inherent in random matrix statistics. While the Harer-Zagier formula originates from the study of the moduli space of curves, it also gives rise to a fundamental formula in the study of spectral moments of classical random matrices. In this talk, I will introduce the Harer-Zagier type formulas for the classical Hermitian Gaussian random matrix ensembles and present recent results on these formulas for the non-Hermitian random matrix model called the elliptic Ginibre ensemble.

Abstract: In this talk I will show how to derive formulas for the large N expansion of the generating function of connected correlators of the β-deformed Gaussian and Wishart-Laguerre matrix models. I will show that these formulas satisfy the known transformation properties under the exchange of β with 1/β and, using Virasoro constraints, I will derive a recursion formula for the coefficients of the expansion. In the undeformed limit β=1, these coefficients are integers and they have the combinatorial interpretation of generalized Catalan numbers. For generic β, we are led to define the notion of higher genus Catalan polynomials whose coefficients are integer numbers. Based on arXiv:2310.05626.

Abstract: Non-normal matrices are much more sensitive to additive perturbations as compared to their normal counterparts. This sensitivity can be quantified appealing to the so-called self-overlap of their left and right eigenvectors. We study the first moment of the self-overlap for complex eigenvalues of N × N matrices in the real Ginibre ensemble. As part of this talk, we provide an overview of the techniques pertinent to our studies, such as incomplete Schur decomposition and Grassmann integration. The derived expression for the first moment is valid for finite size N of matrices, however we are mainly interested in its large-N asymptotic behaviour. This is calculated in three different regions of the complex plane with different density of eigenvalues: the spectral bulk, the spectral edge and a region of eigenvalue depletion close to the real line. New results in this talk are compared to existing results for the complex Ginibre ensemble and verified using numerical simulation.

Abstract: An elementary derivation of the Borodin-Sinclair-Forrester-Nagao Pfaffian point process, which characterises the law of real eigenvalues for the real Ginibre ensemble in the large matrix size limit, uses the averages of products of characteristic polynomials. This derivation reveals a number of interesting structures associated with the real Ginibre ensemble such as the hidden symplectic symmetry of the statistics of real eigenvalues and an integral representation for the $K$-point correlation function for any $K\in \N$ in terms of an asymptotically exact integral over the symmetric space $U(2K)/USp(2K)$.

Abstract: In this talk, I will discuss a universal phenomenon that arises from the random matrix theory, known as 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. Our proof relies on a Riemann-Hilbert (RH) characterization of the Bessel kernel. This is based on the joint work with Lun Zhang and can be found on arXiv:2309.06733.

Abstract: This talk introduces techniques for using the large deviations of interacting particle systems to study the large-N asymptotics of generalized Bessel functions. These functions arise from a versatile approach to special functions known as Dunkl theory, and they include as special cases most of the spherical integrals that have captured the attention of random matrix theorists for more than two decades. I will give a brief introduction to Dunkl theory and then present a result on the large-N limits of generalized Bessel functions, which unifies several results on spherical integrals in the random matrix theory literature. These limits follow from a large deviations principle for radial Dunkl processes, which are generalizations of Dyson Brownian motion. If time allows, I will discuss some further results on large deviations of radial Heckman-Opdam processes and/or applications to asymptotic representation theory. Joint work with Jiaoyang Huang.

Abstract: We consider two classical and extensively studied models of random objects: eigenvalues of a GUE random matrix and random integer partitions distributed according to the Schur measure. We express the largest element of these random sets as maxima of independent random variables. We then proceed to rescale the largest eigenvalue of the GUEN written as a maximum of N independent random variables with the classical Poisson approximation for sums of indicators. We use for this the Okamoto-Noumi-Yamada theory of the sigma-form of the Painlevé equation applied to random matrix theory by Forrester-Witte (we will recall part of this theory). By doing so, we find a new expression for the cumulative distribution function of the GUE Tracy-Widom distribution which is shown to be equivalent to the classical one using manipulations à la Forrester-Witte. Last, we will show that the Tracy-Widom distribution is also a maximum of an infinite umber of independent random variables.

Abstract: The quantum transport problem for $1$ dimensional disordered wires can be modeled by the Dorokhov-Mello-Pereyra Kumar (DMPK) equation that is similar to the Dyson Brownian motion, and if the time-reversal symmetry is broken, the DMPK equation has a free fermion solution, which is, after taking the metallic limit, a biorthogonal ensemble. The biorthogonal ensemble has the form $\prod_{1 \leq i < j \leq n} (x_i - x_j)(f(x_i) - f(x_j)) \prod^n_{i =1} x^{\alpha}_i e^{-nV(x_i)}, \quad f(x) = \sinh^2(\sqrt{x}).$ It is a determinantal point process, and the correlation kernel can be expressed by biorthogonal polynomials. In this talk we discuss an approach to the Plancherel-Rotach type asymptotics of the biorthogonal polynomials by vector Riemann-Hilbert problems. 

Abstract: In this talk, I will present new results on the disk counting statistics of the eigenvalues of truncated unitary matrices. More specifically, if T is the upper-left submatrix of a Haar distributed unitary matrix of size (n+k) x (n+k), we prove that as n tends to infinity with k fixed, the associated moment generating function verifies asymptotics of the form exp(C_1 n +C_2 + o(1)), where the constants C_1 and C_2 are given in terms of the incomplete Gamma function. Our proof relies on the uniform asymptotics of the incomplete Beta function. This talk is based on a joint work with Christophe Charlier and Yacin Ameur.

Abstract: In this talk, I will focus on the analysis of a specific class of random matrices called randomly segmented tridiagonal quasi-Toeplitz (rstq-T) matrices. These matrices exhibit a distinctive structure and arise in various contexts in physics. I encountered them during my study of chromatin dynamics in live cells, where I modeled a Rouse polymer embedded in random phase-separated liquid droplets.

Abstract: We will show that, under a condition on the Schatten p norms (p<1) of the Toeplitz operator associated with the kernel of a DPP, whose symbol is the indicator function of a domain, the bipartite entanglement entropy is proportional to the variance. This leads to equivalences between hyperuniformity classes and classes of growth for the entanglement entropy (area law and area law with log correction). Examples include the fermionic model in several dimensions considered by Gioev and Klich (PRL, 2006), which is a multidimensional version of the sine DPP, where a log correction to the area law shows up; the fermionic model on Riemann surfaces of Charles and Estienne (CMP, 2020, a work which strongly influenced this research), the infinite Ginibre process and its polyanalytic versions in higher Landau levels, which belong to the large class of Weyl-Heisenberg ensembles, a DPP defined via the action of the Heisenberg group, dependent on a function g (for choices of g within Hermite functions we are led to the mentioned Ginibre-type ensembles). These last classes of DPPs enjoy an area law as a consequence of their Class I hyperuniformity.

Abstract: I will introduce the first random matrix model of a complex beta ensemble. The main feature of the model is that the exponent beta of the Vandermonde determinant in the joint probability density function (j.p.d.f) of the eigenvalues can take any real positive number. However, when beta=2, the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors. This is joint work with Francesco Mezzadri and can be found on arXiv:2305.13184. 

Abstract: We discuss systematic symmetry classification of Lindblad superoperators describing general (interacting) open quantum systems coupled to a Markovian environment. Our classification is based on the behavior of the Lindbladian under antiunitary symmetries and unitary involutions. We find that Hermiticity preservation reduces the number of symmetry classes, while trace preservation and complete positivity do not, and that the set of admissible classes depends on the presence of additional unitary symmetries: in their absence or in symmetry sectors containing steady states, Lindbladians belong to one of ten non-Hermitian symmetry classes; if however, there are additional symmetries and we consider non-steady-state sectors, they belong to a different set of 19 classes. In both cases, it does not include classes with Kramer’s degeneracy. While the abstract classification is completely general, we then apply it to spin-1/2 chains. We explicitly build examples in all ten classes of Lindbladians in steady-state sectors, describing standard physical processes such as dephasing, spin injection and absorption, and incoherent hopping, thus illustrating the relevance of our classification for practical physics applications. Finally, we show that the examples in each class display unique random-matrix correlations. To fully resolve all symmetries, we employ the combined analysis of bulk complex spacing ratios and the overlap of eigenvector pairs related by symmetry operations.

Abstract: We consider a complex Ginibre ensemble of random matrices with a deformation $H=H_0+A$, where $H_0$ is a Gaussian complex Ginibre matrix and $A$ is a rather general deformation matrix. The analysis of such ensemble is motivated by many problems of random matrix theory and its applications. We use the Grassmann integration methods to obtain integral representation of spectral correlation functions of the first and the second order and discuss the analysis of these representations with a saddle point method. Applications of such an analysis to the problems of local regime of the deformed Ginibre ensemble will be discussed. 

Abstract: The Jacobi Unitary Ensemble (JUE) is a unitarily-invariant matrix model which admits various relevant applications. One way to describe its joint law is by taking the radial part of the compression by two orthogonal projections of a Haar-distributed unitary matrix. This realization has proved useful in understanding the JUE in statistical applications. To extend these results to the dynamical setting, the hermitian Jacobi process was introduced by replacing the Haar unitary matrix with a unitary Brownian motion. Despite the explicit knowledge of joint law of its eigenvalues, determining its large-size limit (in the sense of *-distribution) is notoriously difficult compared to the JUE ensemble. In this talk, we will explore the spectral dynamics of the hermitian Jacobi process in both finite and infinite dimensional settings. We will discuss different approaches for computing its moments and determining the spectral distribution of its large-size limit. Additionally, we will investigate dynamical analogues for some results of the JUE and their applications to quantum information theory. 

Abstract: Universality of Hermitian random matrix theory (RMT) has turned out to appear in various isolated quantum systems, such as quantum chaotic systems and many-body delocalized systems. Recently, open quantum many-body systems attached to an external environment has attracted much attention due to the development of experimental techniques. While non-Hermitian RMT should play a crucial role in understanding such open quantum many-body systems, the fundamental property of the non-Hermitian RMT, i.e., universality classes of the level-spacing distributions and their relation to symmetries, was not identified. In this talk, we discover two additional universal classes caused by the transposition symmetry [1] besides the previously known universality class by Ginibre. We numerically and semi-analytically discuss that there are only three universality classes for the level-spacing distribution in the bulk of the spectrum, while there are 38 different symmetry classes. We show that the newly found universality classes indeed manifest themselves in dissipative quantum many-body systems described by non-Hermitian Hamiltonians or the Lindblad equation, while the universality breaks down due to localization [2,3]. 

[1] R.H., K. Kawabata, N. Kura and M. Ueda., Phys. Rev. Research 2 (2), 023286 (2020). 

[2] R.H., K. Kawabata and M. Ueda., Phys. Rev. Lett. 123, 090603 (2019). 

[3] R.H., M. Nakagawa, T. Haga and M. Ueda., arXiv:2206.02984 (2022). 

Abstract: In this talk, I will discuss two-dimensional Coulomb gases at a specific temperature with free or Neumann boundary conditions, which can be realized as eigenvalues of random normal matrices or planar symplectic ensembles. I will present the asymptotic expansion of the partition functions when the underlying field is radially symmetric. Notably, our findings stress that the expansion contains topological data of the associated droplet. This is based on joint work with Nam-Gyu Kang and Seong-Mi Seo.

Abstract: We study real eigenvalues of 2n x 2n real elliptic Ginibre matrices indexed by a non-Hermiticity parameter 0\leq \tau<1, in both the strong and weak nonHermiticity regime. In both regimes, we prove a Central Limit Theorem for the number of real eigenvalues. We also find the asymptotic behaviour of the probability p_{2n,2k}(\tau) that exactly 2k eigenvalues are real. In the strong non-Hermiticity regime, where \tau is fixed, we find that n^{-1/2} \log p_{2n,2k}(\tau) has a limit (depending on \tau). This result is an extension of the result for \tau = 0 by Kanzieper, Poplavski, Timm, Tribe and Zabronski. In the weak non-Hermiticity regime we obtain an upper bound for this probability. This is joint work with Sung-Soo Byun and Nick Simm.

Abstract: We consider random matrix ensembles on the Hermitian matrices that are heavy tailed, in particular not all moments exist, and that are invariant under the conjugate action of the unitary group. The latter property entails that the eigenvectors are Haar distributed and, therefore, factorise from the eigenvalue statistics. We prove a classification for stable matrix ensembles of this kind of matrices with the help of the classification of the multivariate stable distributions and the harmonic analysis on symmetric matrix spaces. They can be classified by the stability parameter and the spectral measure, apart from a scaling and a shift. Moreover, we address the question how these ensembles can be generated from the knowledge of the first two quantities. We consider a sum of a specific construction of identically and independently distributed random matrices that are based on Haar distributed unitary matrices and stable random vectors. For this construction, we derive the rate of convergence in the supremum norm and show that this rate is optimal. As a consequence we also give the rate of convergence in the total variation distance. 

Abstract: We discuss systematic symmetry classification of Lindblad superoperators describing general (interacting) open quantum systems coupled to a Markovian environment. Our classification is based on the behavior of the Lindbladian under antiunitary symmetries and unitary involutions. We find that Hermiticity preservation reduces the number of symmetry classes, while trace preservation and complete positivity do not, and that the set of admissible classes depends on the presence of additional unitary symmetries: in their absence or in symmetry sectors containing steady states, Lindbladians belong to one of ten non-Hermitian symmetry classes; if however, there are additional symmetries and we consider non-steady-state sectors, they belong to a different set of 19 classes. In both cases, it does not include classes with Kramer’s degeneracy. While the abstract classification is completely general, we then apply it to spin-1/2 chains. We explicitly build examples in all ten classes of Lindbladians in steady-state sectors, describing standard physical processes such as dephasing, spin injection and absorption, and incoherent hopping, thus illustrating the relevance of our classification for practical physics applications. Finally, we show that the examples in each class display unique random-matrix correlations. To fully resolve all symmetries, we employ the combined analysis of bulk complex spacing ratios and the overlap of eigenvector pairs related by symmetry operations.

Abstract: I will talk about the multivariate linear statistics (also called U-statistics) of the determinantal point processes on the unit spheres. I will first present a graphical representation for the cumulants of the multivarite linear statistics, then I will briefly explain how we get the first and 2nd Wiener chaos from this graphical representation and their correspondences. The method can be applied to any other determinantal point process. This is based on the joint work with F. Goetze and D. Yao.

Abstract: The β-ensemble involves a joint density of energy levels where β can be understood as the inverse temperature of an equivalent Coulomb gas model. Dumitriu and Edelmen proposed a matrix representation of β-ensemble. If we consider β = N^{−γ} then we find that the Anderson localization-delocalization transition occurs at γ = 1 and ergodicity breaks down at γ = 0. Thus Non-Ergodic Extended (NEE) phase is observed over a finite interval of parameter values (0 < γ < 1). We find that the level repulsion-clustering transition coincides with the breaking of ergodicity while long-range correlation among the energy levels exhibit a heterogeneous behavior in the NEE regime. We also observe that there are O(N^γ) localized states in the NEE regime which indicates a possibility of spectral inhomogeneity. [Based on the work under the supervision of Anandamohan Ghosh (10.1103/PhysRevE.105.054121).]

Abstract: We consider determinantal Coulomb gas ensembles with a class of discrete rotational symmetric potentials whose droplets consist of several disconnected components. Under the insertion of a point charge at the origin, we derive the asymptotic behaviour of the correlation kernels both in the macro- and microscopic scales. In the macroscopic scale, this particularly shows that there are strong correlations among the particles on the boundary of the droplets. In the microscopic scale, this establishes the edge universality. For the proofs, we use the nonlinear steepest descent method on the matrix Riemann-Hilbert problem to derive the asymptotic behaviours of the associated planar orthogonal polynomials and their norms up to the first subleading terms. Based on the joint work with Sung-Soo Byun. (arXiv:2210.04019).

Abstract: We consider a uniform random Hermitian matrix acting on the tensor product of two finite-dimensional vector spaces, and study the probability distribution of its partial traces (marginals). More specifically, we find formulas for mixed moments of matrix entries of partial traces. Our main tool is Weingarten calculus, which provides a method for computing integrals of polynomial functions with respect to Haar measure on the unitary group. This talk is based on joint work with Colin McSwiggen (arXiv:2210.11349).

Abstract: We study the statistical behaviour of entanglement in quantum bipartite systems over fermionic Gaussian states as measured by von Neumann entropy. The average von Neumann entropy formulas with and without particle number constrain have been recently obtained, whereas the main results of this work are the exact yet explicit formulas of the corresponding variances. In particular, the results resolve the recent conjecture on the variance in the case of no particle number constrain. Different than the previous methods in computing the exact variances in other generic state models, the key ingredient in proving the results of this work relies on a new simplification framework. The new framework consists of a set of new tools of what we refer to as dummy summation and re-summation techniques in simplifying finite summations. As a byproduct, the proposed framework leads to various new transformation formulas of hypergeometric functions.