2021 Seminars

Abstract: In this talk, I will discuss the statistical properties of entanglement entropy, which serves as a natural measure of quantum correlations between a subsystem and its complement. Entanglement is a defining feature of quantum theory and understanding its statistical properties has applications in many areas of physics (quantum information, statistical mechanics, condensed matter physics, black hole thermodynamics).

First, I will introduce the physical model and explain its relevance for practical applications. Second, I will explain how the statistical ensemble of quantum states can naturally be described through the methods of random matrix theory. Third and finally, I review a number of new results describing the typical properties (e.g., average, variance) of the entanglement entropy for various ensembles of quantum states (general vs. Gaussian, arbitrary vs. fixed particle number).

Abstract: The “color-flavor transformation”, conceived as a kind of generalized Hubbard-Stratonovich transformation, is a variant of the Wegner-Efetov supermatrix method for disordered electron systems. Tailored to quantum systems with disorder distributed according to the Haar measure of a compact Lie group of any classical type (A, B, C, or D), it has been applied to Dyson’s Circular Ensembles, random-link network models, quantum chaotic graphs, disordered Floquet dynamics, and more. We review the method and, in particular, explore its limits of validity. We also sketch a new alternative method to treat models where the color-flavor transformation fails.

Abstract: This is a continuation of a talk from last week that was cut short due to technical difficulties. The abstract is below.

When the initial data of a discrete integrable system is sampled according to a probability measure, the Lax matrix of the system becomes a random matrix. The goal is to study the spectrum of random Lax matrices of integrable systems. In this setting we consider the discrete defocusing nonlinear Schrödinger equation in its integrable version, that is called Ablowitz Ladik lattice. When the initial data is sampled from the Gibbs ensemble the Lax matrix of the Ablowitz Ladik lattice turns into a random matrix that is related to the circular beta-ensemble at high temperature. We obtain the density of states of the random Lax matrix, when the size of the matrix goes to infinity, by establishing a mapping to the one-dimensional log-gas. The density of states is obtained via a particular solution of the double confluent Heun equation. Joint work with Guido Mazzuca.

Abstract: When the initial data of a discrete integrable system is sampled according to a probability measure, the Lax matrix of the system becomes a random matrix. The goal is to study the spectrum of random Lax matrices of integrable systems. In this setting we consider the discrete defocusing nonlinear Schrödinger equation in its integrable version, that is called Ablowitz Ladik lattice. When the initial data is sampled from the Gibbs ensemble the Lax matrix of the Ablowitz Ladik lattice turns into a random matrix that is related to the circular beta-ensemble at high temperature. We obtain the density of states of the random Lax matrix, when the size of the matrix goes to infinity, by establishing a mapping to the one-dimensional log-gas. The density of states is obtained via a particular solution of the double confluent Heun equation. Joint work with Guido Mazzuca.

Abstract: Gaussian beta ensembles are generalizations of the GOE, GUE and GSE in terms of the joint density of the eigenvalues. When the parameter β is fixed, their empirical distribution converges weakly to the semicircle distribution, almost surely, which is called Wigner’s semicircle law. Gaussian fluctuations around the semicircle distribution are also well-studied: see Johansson (1998) for an approach based on analyzing the joint density, Dumitriu and Edelman (2006) for an approach using a random tridiagonal matrix model, and Cabanal-Duvillard (2001) for a dynamical approach. What happens when the parameter β varies as the system size N tends to infinity? It turns out that Wigner’s semicircle law holds as long as βN tends to infinity. In this regime, Gaussian fluctuations are almost the same as those in the case β is fixed. When βN stays bounded, referred to as a high temperature regime, the limiting distribution belongs to a family of probability measures of associated Hermite polynomials, see Allez et al. (2012) and Duy and Shirai (2015). In a high temperature regime, Gaussian fluctuations around the limit were established by using a random tridiagonal matrix model; see Nakano and Trinh (2016) or Trinh (2019). This talk introduces a dynamical approach to study Gaussian fluctuations with further relations with associated Hermite polynomials.

Abstract: Random matrix theory has played an important role in the understanding of the spectral fluctuations of various physical systems. In these studies, the tools that are widely used include the distribution of spacing, spacing ratios, number variance, etc. In this talk, I will present recent developments based on the higher-order spacing ratios. Results obtained using the circular ensembles and their superposed spectra will be presented. Finally, their applications to the various complex physical systems will be demonstrated.

Abstract: Consider a finite product of real asymmetric N x N random matrices with i.i.d. Gaussian entries. Recently, Forrester and Ipsen obtained finite-N formulae for the correlation functions of the real eigenvalues of such products in terms of a Pfaffian point process. I will discussed recent work where we obtain asymptotic estimates for the correlation kernel of the process as N tends to infinity, in particular establishing universality of the real eigenvalues in the bulk and spectral edge regimes. I will explain how we apply such estimates to prove central limit theorems for linear statistics and to establish universality of the largest real eigenvalue.

Joint work with Will FitzGerald (University of Manchester).

Abstract: In this talk, I will discuss complex eigenvalues of non-Hermitian random matrices with symplectic symmetry, which are known to form Pfaffian point processes. In particular, I will present various scaling limits of symplectic ensembles and explain their unified integrable structure of Wronskian form. Examples include edge scaling limits of the Ginibre ensemble (with boundary confinements) and bulk/edge scaling limits of the elliptic Ginibre ensemble in the almost-Hermitian regime.

Beyond standard universality classes, I will also introduce scaling limits of the Mittag-Leffler ensemble at the singularity. Furthermore, for symplectic ensembles with general external potentials, I will present the characterization of translation invariant scaling limits by virtue of rescaled mass-one and Ward’s equations.

Abstract: Heavy-tailed random matrices have surprising and novel effects that can be hardly seen with the classical ensembles. For instance, in recent years it was shown that heavy-tailed Wigner matrices can exhibit localised eigenvector statistics for the eigenvalues in the tail while everything stays the same as we know it for the bulk statistics of a GUE. This effect, some intriguing as well as real world applications, and some own numerical experiments have motivated us to study invariant heavy tailed random matrices. One of the questions we have addressed has been about central limit theorems at fixed matrix dimensions and invariant random matrices that are stable when adding independent copies of the random matrix under consideration. I will report on our new findings and will sketch the main ideas of their proofs in the present talk.

These projects have been carried out in collaboration with Jiyuan Zhang and Adam Monteleone.

Abstract: Consider an N by N matrix X of complex entries with iid real and imaginary parts. We show that the local density of eigenvalues of X*X converges to the Marchenko-Pastur law on the optimal scale with probability 1. We also obtain rigidity of the eigenvalues in the bulk and near both hard and soft edges. Here we avoid logarithmic and polynomial corrections by working directly with high powers of expectation of the Stieltjes transforms. We work under two sets of assumptions: either the entries have bounded moments or the entries have a finite 4th moment and are truncated at N^(1/4). In this work we simplify and adapt the methods from prior papers of Götze-Tikhomirov and Cacciapuoti-Maltsev-Schlein to covariance matrices. This is joint work with Anastasis Kafetzopoulos.

Abstract: Consider a two-dimensional Coulomb droplet. It is expected that different charges at the edge should be correlated in a relatively strong way. The physical picture is that the screening cloud about a charge at the boundary has a non-zero dipole moment, which gives rise to a slow decay of the correlation function. This phenomenon was studied (on the “physical” level of rigor) by Forrester and Jancovici in a paper from 1995 for the elliptic Ginibre ensemble. Coincidentally, a recent joint work between myself and Joakim Cronvall on reproducing kernels turned out to be closely related to this question.

 Indeed, if there are n particles, we obtain that the order of magnitude of the correlation function K_n(z,w) is proportional to √n if z,w are on the boundary and z ≠ w, while K_n(z,z) is proportional to n. This gives the “slow decay” of correlations at the boundary. (For comparison, if one of the charges (say z) is in the bulk, then K_n(z,w) decays quickly for z ≠ w: |K_n(z,w)| ≲ exp(-c√n).)

 In addition, we find that in the limit as n → ∞, there emerges the following correlation kernel K(z,w) for z,w on the (outer) boundary:

(0.1)    K(z,w)=(2π)^(-1/2)(ΔQ(z)ΔQ(w))^(1/4) [√φ’(z)(√φ’(w))*]/[φ(z)(φ(w))*-1]

Here, we assumed (for simplicity) that the droplet is connected and that z,w are on the outer boundary curve Γ; then φ is a Riemann mapping from Ext(Γ) to the exterior disk {|z| > 1}. (Thus, it should be understood that |φ(z)| = |φ(w)| = 1 in (0.1).) Finally, Q is the (rather arbitrary) external potential used to define the ensemble. For example: Q(x+iy) = ax^2 + by^2 in the case of elliptic Ginibre.

The kernel S(z,w) given by the ratio of square brackets in (0.1) can be recognised as the so-called Szegő kernel of the boundary curve Γ. (That S(z,z) = ∞ reflects the fact that long-range vs. short-range interactions take place on different scales.)

Our method for deriving these results builds on the technique of full-plane orthogonal polynomials due to Hedenmalm and Wennman (work to appear in Acta Math). Using summation by parts and “tail-kernel approximation”, we in fact obtain certain asymptotic results for the canonical correlation kernel in cases beyond the boundary-boundary case; in particular, our results extend nicely to the exterior of the droplet.

In the basic case of the Ginibre ensemble, we obtain more precise asymptotics for K_n(z,w) (an expansion in powers of 1/n) by developing techniques found in Szegő’s classical work on the distribution of zeros of partial sums s_n(z) = 1+z+…+z^n/n! (n → ∞).

Ameur, Y., Cronvall, J., Szegő type asymptotics for the reproducing kernel in spaces of full-plane weighted polynomials. Arxiv: 2107.11148.

(We are planning an update in a relatively near future, and we are therefore particularly grateful for comments.)

Abstract: Non-Hermitian random matrices with symplectic geometry provide examples for Pfaffian point processes in the complex plane. They are characterised by a matrix-valued kernel of skew-orthogonal polynomials, however finding an appropriate set of polynomials for a given matrix ensemble is difficult.

In this talk I will present an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation. Additionally, for the symplectic elliptic Ginibre ensemble I will show how to compute the microscopic large-N limit of the kernel at the origin and prove its universality.

This is based on joint work with Gernot Akemann and Iván Parra. If time permits, I will also mention some results from an ongoing project with Sung-Soo Byun.

Abstract: In this talk, I will revisit the problem of the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing a novel approach which combines field theoretical, probabilistic and integrable techniques. My goal will be to expand the program of the weak noise theory, which maps the large deviations onto a non-linear hydrodynamic problem, and to unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system.

Abstract: Spectra of asymmetric random matrices have been studied for decades. Despite this, densities of real eigenvalues are not yet fully understood. In particular, their large-N form is obtained through elaborate asymptotic analysis of the finite size results. This is in strong contrast with the complex densities, for which various tools like Feynman diagrams, cavity equations and free probability provide large-N formulas without the need to solve finite size models.

Abstract: In this talk, I will discuss the real eigenvalues of the real elliptic Ginibre matrix, the model which provides a natural bridge between Hermitian and non-Hermitian random matrix theories. In the almost-Hermitian regime pioneered by Fyodorov, Khoruzhenko and Sommers, I will present the large-N expansion of the mean and the variance of the number of real eigenvalues. Furthermore I will explain the limiting empirical spectral distributions of the real eigenvalues which interpolate the Wigner semicircle law and the uniform distribution. The proofs are based on the skew-orthogonal representation of the correlation kernel due to Forrester and Nagao.

Abstract: Especially after Voiculescu’s groundbreaking discovery of the phenomenon of asymptotic freeness, it has become a standard technique to use Hilbert space operators in order to describe the behaviour of random matrices when their size tends to infinity. Whenever possible, this is done in such a way that the asymptotic eigenvalue distribution of any hermitian polynomial evaluated in the considered random matrices converges in an appropriate sense towards the spectral distribution of the same polynomial evaluated in the limiting operators. The latter object can be studied by operator algebraic means, in particular using tools from free probability.

When involving inverses, i.e., when passing from polynomials to rational functions, the situation becomes more delicate. Indeed, not only the convergence can break down but already the evaluation might fail to exist. In my talk, which is based on joint work with Benoît Collins, Akihiro Miyagawa, Félix Parraud, and Sheng Yin, I will present some general approach to these problems, hereby answering a question of Roland Speicher.

Abstract: The standard approach to dynamical random matrix models relies on the description of trajectories of eigenvalues. Using the analogy from optics, based on the duality between the Fermat principle (rays) and Huygens principle (wavefronts), we formulate  the Hamilton-Jacobi dynamics for large random matrix models, and we relate this dynamics to Voiculescu equation. The resulting formalism describes a broad class of random matrix models in a unified way, including normal (Hermitian or unitary) as well as strictly non-normal dynamics. HF formalism applied to Brownian bridge dynamics allows one for calculations of the asymptotics of the Harish-Chanda-Itzykson-Zuber integrals.

Abstract: Muttalib-Borodin determinants are generalizations of Hankel determinants and are relevant in the study of point processes known as Muttalib-Borodin ensembles. The notable feature of Muttalib-Borodin ensembles is that neighboring points x_j, x_k repel each other as ~(x_k - x_j)(x_k^θ-x_j^θ), which differs, for θ ≠ 1, from the simpler and more standard situation ~(x_k - x_j)^2. In this talk I will discuss some recent results on asymptotics for Muttalib-Borodin determinants with Fisher-Hartwig singularities and I will present some applications.

Abstract: In random matrix theory, the question of large deviations of spectral quantities (that is: how does the probability that these quantities take atypical values decay?) remains mysterious outside of some specific models. However, recent advances on this question make use of HCIZ integrals (also known as spherical integrals) as proxy for the largest eigenvalues. In this talk I will expose how to determine the asymptotics of these spherical integrals when the rank is constant and I will explain how to use these integrals to estimate the large deviations of the largest eigenvalues. This talk is mainly based on two joint works with A. Guionnet and a joint work with F. Augeri and A. Guionnet.

Abstract: Spherical integrals as the HCIZ integral can be thought as Fourier transforms in RMT, and their asymptotics useful to capture the probability of rare events in random matrix theory. In this talk, I will discuss how to study the asymptotics of the HCIZ integral and the rectangular spherical integral by studying large deviations for Dyson Brownian motion and how this can be used to study the probability that the spectral measure of the matrix A+UBU* takes an unexpected value. This talk is based on a old work with O. Zeitouni, and recent work with S. Belinschi and J. Huang as well as a work in progress with J. Huang.

Abstract: Many fields of modern sciences are faced with high-dimensional data sets. In this talk, we investigate the spectral properties of large sample correlation matrices.

First, we consider a p-dimensional population with iid coordinates in the domain of attraction of a stable distribution with index α ϵ (0,2). Since the variance is infinite, the sample covariance matrix based on a sample of size n from the population is not well behaved and it is of interest to use instead the sample correlation matrix R. We find the limiting distributions of the eigenvalues of R when both the dimension p and the sample size n grow to infinity such that p/n → γ. The moments of the limiting distributions H_{α,γ} are fully identified as the sum of two contributions: the first from the classical Marchenko-Pastur law and a second due to heavy tails. Moreover, the family { H_{α,γ} } has continuous extensions at the boundaries α = 2 and α = 0 leading to the Marchenko-Pastur law and a modified Poisson distribution, respectively. A simulation study on these limiting distributions is also provided for comparison with the Marchenko-Pastur law.

In the second part of this talk, we assume that the coordinates of the p-dimensional population are dependent and p/n ≤ 1. Under a finite fourth moment condition on the entries we find that the log determinant of the sample correlation matrix R satisfies a central limit theorem. In the iid case, it turns out the central limit theorem holds as long as the coordinates are in the domain of attraction of a stable distribution with index α > 3, from which we conjecture a promising and robust test statistic for heavy-tailed high-dimensional data. The findings are applied to independence testing and to the volume of random simplices.

Abstract: We will talk about a way to construct an “effective statistical field theory” for fixed points in “generic” complex systems based on rotational symmetry. Rather than focus on particular technical details, we will attempt to give a greater overview of this framework and its connection to Random Matrix Theory through discussions of several known results as well as several open questions. We will talk about non-autonomy, non-linearity, and anisotropy and effects on stability and resilience.

Abstract: Radio channel modeling and system performance analysis for wireless communications exploit random matrix theory since the very introduction of the multi-antenna transmission paradigm in late ’90’s. Matrices from polynomial ensembles have been providing realistic and analytically handy models for communications taking place at frequencies typical of the third and fourth generation of mobile telephony, and still serve to analyze low-frequency performance in the most recent and currently developing fifth generation.

The shift to far-higher frequencies planned for fifth and, especially, sixth generation of mobile communications is leading to the adoption of radio channel models putting emphasis on geometry-related rather than on scattering-related information.

In this scenario, both Euclidean as well as random Vandermonde matrices play a major role.

The talk focuses on a representative case of 6G multi-antenna link, where signal transmission from transmit to receive uniform linear array is aided by a so-called “reflective intelligent surface”, a passive device with signal-bearing capabilities only. This involves the analysis of products of random Vandermonde matrices with complex entries of unit modulus, with either random or deterministic matrices interspersed.

Abstract: Recently there has been a wave of research into products of real asymmetric random matrices. Because these random matrices are asymmetric, they have both real and complex eigenvalues, with the number of each being random. The real eigenvalues of an asymmetric random matrix interact in an interesting way with taking products, in particular, longer products tend to lead to more real eigenvalues. We look at a particular ensemble, products of so-called “truncated orthogonal” matrices and prove a conjecture about the number of real eigenvalues and their distribution along the real line. Proving this conjecture amounted to a problem in asymptotic analysis, and I will go over the key tricks we used to carry this out.

This was joint work with Francesco Mezzadri (Bristol) and Nicholas Simm (Sussex). Our paper can be found here: https://arxiv.org/abs/2102.08842

Abstract: We analyze spectral properties of generic quantum operations, which describe open systems under assumption of a strong decoherence and a strong coupling with an environment. In the case of discrete maps, the spectrum of a quantum stochastic map displays a universal behaviour: it contains the leading eigenvalue λ_1 = 1, while all other eigenvalues are restricted to the disk of radius R < 1. Similar properties are exhibited by spectra of their classical counterparts - random stochastic matrices.

In the case of a generic dynamics in continuous time, we introduce an ensemble of random Lindblad operators, which generate Markov evolution in the space of density matrices of a fixed size. Universal spectral features of such operators, including the lemon-like shape of the spectrum in the complex plane, are explained with a non-hermitian random matrix model. The structure of the spectrum determines the transient behaviour of the quantum system and the convergence of the dynamics towards the generically unique invariant state. The quantum-to-classical transition for this model is also studied and the spectra of random Kolmogorov operators are investigated.

Abstract: Complex systems are often non-stationary, typical indicators are continuously changing statistical properties of time series. In particular, the correlations between different time series fluctuate. Models that describe the multivariate amplitude distributions of such systems are of considerable interest. We view a set of measured, non-stationary correlation matrices as an ensemble for which we set up a random matrix model. We use this ensemble to average the stationary multivariate amplitude distributions measured on short time scales and thus obtain for large time scales multivariate amplitude distributions which feature heavy tails. We explicitly work out four cases, combining Gaussian and algebraic distributions. For the latter we use a determinantal generalization of the Wishart distribution, known as matrix variate t distribution. We also calculate its first and second matrix moments.

In summary, we provide, first, explicit multivariate distributions for non-stationary complex systems and, second, a tool that quantitatively captures the degree of non-stationarity in the correlations. We present some first applications to financial data.

Abstract: The Muttalib-Borodin ensemble, with potential function V, is a log-gas model with two types of interaction, and its joint probability density function is proportional to

Π_{1≤i<j≤n} (x_i - x_j) (x_i^θ - x_j^θ) Π_{1≤i≤n} x_i^α exp[-n V(x_i)].

When V(x) = x, interesting limiting distribution is found for the smallest particles. In this talk we discuss our recent result on the universality of the Muttalib-Borodin ensemble at the hard edge, when θ is an integer, that is, the limiting distribution is not affected by V.

This is joint work with Lun Zhang.

Abstract: In this talk, we consider a family of tronquée solutions of the Painlevé II equation q''(s) = 2q(s)^3 + sq(s) - (2α + 1/2),    α > -1/2,

which is characterized by the Stokes multipliers s_1 = -exp[-2απi],  s_2 = ω,  s_1 = -exp[2απi] with ω being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if ω = 0. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for α > -1/2 and ω ≥ 0, with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters α and ω are chosen to be special values. Some applications of our results in random matrix theory are also discussed. Joint work with Dan Dai and Shuai-Xia Xu.

Abstract: I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painlevé equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.

Abstract: We will consider a two-dimensional gas with repulsive Coulomb interactions under an external field. More precisely, we will focus on a Coulomb gas system which is determinantal and subject to a volume constraint. In the presence of a hard-wall constraint to change the equilibrium, the density of the equilibrium measure acquires a singular component at the hard wall.

 In this talk, I will discuss the local statistics of Coulomb gases near the hard wall. As the number of particles tends to infinity, the Coulomb gas system properly rescaled at the hard wall converges to a determinantal point process with a kernel expressed as a Laplace-type integral, and this kernel appears in the context of truncated unitary matrices in the regime of weak non-unitarity. I will also explain an approach to universality of the kernel based on a rescaled version of Ward’s equation.

Abstract: I’ll talk about a problem motivated by quantum information theory and raised by Nechita and Pellegrini. They asked to compute the joint distribution of a sub-vector of a random vector in the unit sphere sampled from the heat kernel.

For a single coordinate, the solution is expressed as a bilinear series of Jacobi polynomials and more generally, it involves the so-called Jacobi polynomials in the simplex introduced by Koorwinder. I’ll exhibit the approach used by Nechita and Pellegrini based on pde’s and the one I used to completely solve the problem and based on the decomposition of unitary spherical harmonics under the action of unitary subgroups.

Abstract: Consider a system of N non-intersection Brownian bridges on the time interval [-1,1] such that the first N-m paths start and end at the origin and the m remaining top paths go between arbitrary positions. The Airy process with m wanderers is defined as the motion of these Brownian particles near the edge curve C := {(T,[2N(1-t^2)]^{1/2}) | t ϵ [-1,1]} in the large N limit. In this talk, we focus on the distribution of the maximum of the Airy process with wanderers minus a parabola, which provides a 2m-parameter deformation of the Tracy-Widom GOE distribution. We provide a Fredholm determinant formula for this distribution function. We also discuss the connection with KPZ fluctuations, as well as some results on relations with Painlevé II and other PDEs.

Abstract: We will consider a Coulomb gas consisting of a large number n of identical repelling (logarithmically interacting) point charges, subject to an external field which confines the gas to a finite portion of the plane known as the “droplet”. The statistical properties of the gas depend critically on the inverse temperature β=1/(k_B T).

During my talk I will discuss two recent kinds of results.

The first one makes precise the physical intuition that the gas should with high probability be localized to a small neighbourhood of the droplet. Results of this kind have been known earlier in the case β=1 and for special potentials. (In particular, Brian Rider has given very precise results for the classical Ginibre ensemble.)

The second group of results are valid at low temperatures (β>c log n where n is number of particles) and shows that (under natural assumptions) almost every sample is uniformly separated and equidistributed in the droplet, all the way up to the boundary. These results, which are joint with José-Luis Romero, generalizes and improves on earlier results on the distribution of Fekete-configurations, corresponding to the temperature zero.