2020 Seminars

Abstract: Complex systems, such as neural networks, ecosystems, and the World Wide Web, consist of components that interact along the edges of large, directed graphs. The eigenvalues and right (or left) eigenvectors of the associated adjacency matrix determine the linearised dynamics of dynamical systems in the vicinity of a fixed point.

In this seminar, we will discuss the properties of the leading eigenvalue of the adjacency matrix of random directed graphs with a prescribed distribution of indegrees and outdegrees. The leading eigenvalue of directed graphs has a couple of remarkable properties: The leading eigenvalue of directed graphs is finite even if the maximal outdegree of the graph diverges (contrary to the case of nondirected graphs). In addition, the leading eigenvalue admits a simple analytical expression that only depends on a few parameters of the network. We also determine the inverse participation ratio of the leading right eigenvector and find that there exists a phase transition between a localised and delocalised phase. Remarkably, the phase transition between these two phases is independent of the fluctuations in the indegrees and outdegrees of the graph, and thus exhibits also a high degree of universality.

Abstract: In my talk I will introduce a tridiagonal random matrix model related to the classical Gaussian β-ensemble in the high temperature regime, i.e. when the size N of the matrix tends to infinity with the constraint that βN=2α constant, α>0. I will show how to explicitly compute the mean density of states and the mean spectral measure for this ensemble. Finally, I will apply this result to compute the mean density of states for the periodic Toda lattice in thermal equilibrium.

This talk is based on my recent preprint “On the mean Density of States of some matrices related to the beta ensembles and an application to the Toda lattice”, arXiv preprint:2008.04604, and partly on a joint work with T.Grava, A. Maspero, and A. Ponno “Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit”, Communications in Mathematical Physics, 380 (2020), pp. 811–851. DOI: 10.1007/s00220-020-03866-2.

Abstract: The Gaussian Entire Function (GEF) is a distinguished random Taylor series with independent complex Gaussian coefficients, whose zero set is invariant with respect to isometries of the plane. The topic of this talk is the zero distribution of the GEF, conditioned on the event that no zero lies in a given (large) region. For circular holes Ghosh and Nishry observed that as the radius of the hole tends to infinity, the density of zeros vanishes not only on the given hole, but also on an annulus beyond the (rescaled) hole - a ‘forbidden region’ emerges.

We are concerned with the shape of the forbidden region for general simply connected holes. I plan to discuss how one can study this problem through a type of constrained obstacle problem, and why the forbidden region belongs to a class of algebraic domains - the quadrature domains for subharmonic functions.

Based on joint work with Alon Nishry.

Abstract: We consider the planar orthogonal polynomials {p_n(z)} with respect to the measure supported on the complex plane exp(-N|z|^2)  Π_{j=1,…,ν} |z-a_j|^{2c_j} dA(z), where dA is the Lebesgue measure of the plane, N is a positive constant, {c_1,…,c_ν} are nonzero real numbers greater than -1, and {a_1,…,a_ν} ⊂ D\{0} are distinct points inside the unit disk. The orthogonal polynomials are related to the interacting Coulomb particles with charge +1 for each, in the presence of extra particles with charge +c_j at a_j. For fixed c_j, these can be considered as small perturbations of the Gaussian weight. When ν=1, in the scaling limit n/N=1 and n → ∞, we obtain strong asymptotics of p_n(z) via a matrix Riemann-Hilbert problem. From the asymptotic behavior of p_n(z), we find that, as we vary c_1, the limiting distribution of zeros behaves discontinuously at c_1=0. We observe that the generalized Szegő curve (a kind of potential theoretic skeleton) also behaves discontinuously at c_1=0. We also derive the strong asymptotics of p_n(z) for the case of ν>1 by applying the nonlinear steepest descent method on the matrix Riemann-Hilbert problem of size (ν+1) x (ν+1). This talk is based on joint work with Seung-Yeop Lee.

Abstract: In this talk we will present several recent results on random matrices over the p-adic numbers ℚ_p, and highlight the parallels with usual real/complex random matrix theory which they illustrate. We show that singular numbers (also known as invariant factors or Smith normal forms) of products and corners of random matrices over ℚ_p are governed by the Hall-Littlewood polynomials, in a structurally identical manner to the classical relations between singular values of real/complex random matrices and Heckman-Opdam hypergeometric functions. We use these exact results to analyze the singular numbers of products of many n x n corners of Haar-distributed matrices, showing that for fixed n they obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the large n limit, we find that the analogues of the Lyapunov exponents for such matrix products have equally spaced ‘picket fence’ statistics as in the real/complex case, which are universal within this class of Haar corners. No background on p-adic numbers or p-adic random matrix theory will be assumed.

Abstract: The estimation of covariance matrices plays a fundamental role in portfolio selection in financial engineering. Recently, a new type of estimators of large-dimensional covariance matrices has been introduced. They are called nonlinear shrinkage estimators in financial engineering [1] and rotationally invariant estimators in physics [2]. The construction of the estimators is based on the idea of James-Stein shrinkage and exact relations between eigenvalues of sample covariance matrix and population covariance matrix. I will briefly recall the main ideas behind the construction and sketch how to generalise it to the case of correlated samples.

[1] Olivier Ledoit, and Michael Wolf, Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets Goldilocks, Review of Financial Studies, 30 (2017) 4349-4388.

[2] Joel Bun, Jean-Philippe Bouchaud, Marc Potters, Cleaning large correlation matrices: tools from random matrix theory, Physics Reports, 666 (2017) 1-112.

Abstract: In this talk, we present recent results on local edge fluctuations for sums of unitarily invariant Hermitian matrices, focusing on several regimes where the Airy point process arises. We also discuss some of the ideas behind the result. In particular, we focus on describing a fascinating connection between observables of these models and a special family of lifts of the multivariate Bessel functions, which can be realized as continuous analogues of supersymmetric Schur functions.

Abstract: Interacting particle systems of Calogero-Moser-Sutherland type on ℝ or intervals are described by some root system and coupling constants. From a probabilistic point of view they are multivariate Bessel processes; moreover they are closely related with dynamic random matrix models.

In this talk we discuss some limit results when the coupling constants and/or the number of particles tend to infinity. It turns out that for the limits the behaviour of associated deterministic dynamical systems and the zeroes of classical orthogonal polynomials play a major role.

Abstract: While the method of moments is an effective method of proving the existence of limiting spectral distributions for many models of random matrices, it does not reveal properties of the limiting distribution such as absolute continuity or unimodality.

In this talk we work with random Toeplitz and Hankel matrices and show the absolute continuity of the corresponding limiting distributions. The existence of the limiting distribution was already shown by Hammond-Miller and Bryc-Dembo-Jiang (~2003) and the absolute continuity was settled for the Toeplitz case by Sen and Virag (2011). The result is new for the Hankel matrix.

Our methods also work for Toeplitz and Hankel matrices defined by certain other groups such as ℤ^d. The key idea is to write the random matrix under consideration as a sum of two or more independent random circulant-like matrices.

All this is joint work with Anish Mallick.

Abstract: Dr. Schehr will discuss a system of N one-dimensional free fermions in the presence of a confining trap V(x). For the harmonic trap V(x) ∝ x^2 and at zero temperature, this system is intimately connected to random matrices belonging to the Gaussian Unitary Ensemble (GUE). In particular, the spatial density of fermions has, for large N, a finite support and it is given by the Wigner semi-circular law. Besides, close to the edges of the support, the spatial quantum fluctuations are described by the so-called Airy kernel, which plays an important role in random matrix theory. We will then focus on the joint statistics of the momenta, with a particular focus on the largest one p_{max}. Again, for the harmonic trap, momenta and positions play a symmetric role and hence the joint statistics of momenta is identical to that of the positions. Here we show that novel “momentum edge statistics” emerge when the curvature of the potential vanishes, i.e. for “flat traps” near their minimum, with V(x) ~ x^{2n} and n>1. These are based on generalisations of the Airy kernel that we obtain explicitly. The fluctuations of p_{max} are governed by new universal distributions determined from the n-th member of the second Painlevé hierarchy of non-linear differential equations, with connections to multi-critical random matrix models, which have appeared, in the past, in the string theory literature.

Abstract: 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.

In this talk, we consider the Fredholm determinant det(I-γ K^{Pe}_{s,ρ}), where 0 ≤ γ ≤ 1 and K^{Pe}_{s,ρ} stands for the trace class operator acting on L^2(-s, s) with the classical Pearcey kernel. Based on a steepest descent analysis for a 3 by 3 matrix-valued Riemann-Hilbert problem, we obtain asymptotics of the Fredholm determinant as s → +∞, which is also interpreted as large gap asymptotics in the context of random matrix theory.

This is a joint work with Shuai-Xia Xu and Lun Zhang.

Abstract: Dip-ramp-plateau refers to the graphical shape of the average of the quadratic statistic |Σ_{j=1}^N e^{i k λ_j}|^2, which in turn is closely related to the spectral form factor. This has received prominence in recent studies of the SYK model and many body quantum chaos. For the GUE, a result of Brézin and Hikami from 1997 gives an identity relating this to the cumulative distribution of the density of the LUE with Laguerre parameter a=0. A derivation will be given, due to Okuyama, which can be generalised to show that for the LUE this statistic can be expressed in terms of the cumulative distribution of the density of the JUE with parameter b=0. Some scaling limits will be discussed.

Abstract: Prof. Strahov will explain how the theory of symmetric functions can be applied to product matrix processes with symplectic and orthogonal invariance. These product matrix processes can be understood as scaling limits of Macdonald processes introduced by Borodin and Corwin. The relation with Macdonald processes enables to generalize the recent Kieburg-Kuijlaars-Stivigny formula for products of truncated unitary matrices to symplectic and orthogonal symmetry classes, and to obtain the joint law of squared singular values for products of truncations of Haar distributed symplectic and orthogonal matrices. Based on joint work with Andrew Ahn.

Abstract: Hankel matrices are matrices of moments (see Heine, Handbuch der Kugelfunctionen, 1878) that play a fundamental role in approximation theory. It transpired that the logarithm of the n x n Hankel determinants – depending on parameters in the weight – plays an important role in finite n aspects of integrable systems. The early pioneers are Jimbo, Miwa, Mori, Sato, Ueno, Okamoto, McCoy, Tracy, and Widom.

Since Hankel matrices are moments of positive continuous functions, they form positive definite quadratic forms.

We like to find the smallest eigenvalue, with the aid of polynomials orthogonal with respect to the weight. This talk will focus on the weight characterised by a parameter β>0, w(x) = exp(-x^β),    0 ≤ x < ∞.

Abstract: The SYK model is a random q-fermion interacting model that is large-N solvable. The model is physically interesting because it exhibits chaotic and non-fermi liquid behaviors and possesses a gravity dual description (q≥4). In this talk I will present spectral properties of the SYK model from both random matrix and path integral points of view: the path integral formalism computes leading-order (1/N) properties via saddle point equations efficiently, while the random matrix method readily computes the leading- and the subleading-order (1/N^2) contributions. Interestingly, a new application of RMT is found in the study of the eigenstate entanglement entropy of the SYK model with q=2: the subsystem entanglement Hamiltonian of this random free fermion model belongs to the Jacobi β-ensemble. This is to be compared with the entanglement entropy of a random many-body eigenstate, whose reduced density matrix belongs to the Wishart ensemble. Finally, I explain how these results can be obtained via replica trick in the path integral formalism and discuss the possibility of formulating new RMT questions in the context of quantum systems.

Abstract: We study random matrices X which are the product of a single Ginibre matrix A and a matrix B from a somewhat general ensemble. The mixed moments and eigenvalue distribution of X are given in terms of the moments of powers of the Gram matrix of B. When B is itself a product of Ginibre matrices, we find explicit expressions in terms of Fuss-Catalan numbers. We will discuss motivations from the study of multilayer perceptrons.

Abstract: Non-stationary time series are a fact of life and bad news from the point of view of mathematical analysis of any complex system. A standard procedure is to break up the time horizon the time series contains into shorter epochs in the hope that at least locally a reasonable degree of stationarity can be obtained. By shifting the epochs along the total time horizon, we attempt to obtain some understanding of the time evolution of the system. If, for some complex system several different time series are known, analyzing their correlations is one of the preferred tools available. Yet non-stationarity puts the validity of using a correlation matrix into doubt, and thus almost forces us to time series that are short as compared to the number of time series that we are using if the latter is small. Note that this is not always the case; take neurological data: An EEG produces a limited number of time series roughly ranging from 15 to 30 , but these are very long, say of order of 1000. On the other hand, an fMRI study of the brain produces thousands of time series of length in the hundreds due to the slowness of the measuring process. The latter presents a scenario of short time series.

In financial markets it is usually that “intraday trading” has its own rules, being largely determined by Brownian motion around some trend, that may vary a few times a day. On the other hand, say “end of day” data which are easily available, will to some extent mirror the behaviour in the medium term. In [1] this fact was used to classify the market of the most important stocks of the United States as represented by the companies included in the the SP500 index, using correlation matrices of the returns (%-increase/decrease) in each share. Note that the index itself is a weighted average of these shares. The correlation matrix for epochs were calculated and subject to a noise reduction treatment [2], ordered into clusters using an L1 distance between the matrices and thereafter k-means clustering and a selection scheme based on a branching tree. As we found that the selection of states was somewhat arbitrary and the epochs were a little long for practical use, we reduced the epoch length to 20 trading days; the minimum that seems to be compatible with the remaining noise. We determined an optimal cluster number based on a minimum in the variance we found in the cluster size when applying k-means several times for different values of the noise damping parameter [3]. We also used dimensional scaling to three dimensions mainly to obtain a representation accessible to the eye. In more recent work, partially shown in [4] we further refine the selection of the number of clusters such as to reduce transitions from far away clusters to neighbouring ones to a minimum. We also reduce the shift of the epoch to one day and limit the time horizon to dates in this century in an attempt to improve the practical interest of our work with help of Eduard Seligman, who has ample experience in finance. This in turn allows us to see a trajectory in the dimensionally reduced space, which opens options for a dynamical treatment.

With less intensity we followed two other lines involving eigenvalues. In the first one we use the power-map originally conceived for noise reduction to lift the singularity of the correlation matrix and obtain what we called the emerging spectrum (similar to the null spectrum of chiral matrices) and find plenty of information when plotting e.g. the minimal eigenvalue as a function of the epoch [5, 6]. Another line is to make an en experimental ensemble by choosing at random subsets of the entire set of time series. This seems useful as it improves statistics very much, without distorting the correlation matrices. Applying the idea to the 2-D Ising model with Metropolis dynamics a power law has been calculated and numerically confirmed for a long time-series at the delocalization phase transition [7]. In both cases preliminary applications to market data are available.

References:

[1] Münnix, M. C., Shimada, T., Schäfer, R., Leyvraz, F., Seligman, T. H., Guhr, T., & Stanley, H. E. (2012). Identifying states of a financial market. Scientific reports, 2, 644.

[2] Vinayak, Schäfer R and Seligman T H (2013) Phys. Rev. E 88 032115, Laloux L, Guhr T and Kälber B (2003) J. Phys. A: Math. Gen. 36 3009.

[3] Pharasi, H. K., Sharma, K., Chatterjee, R., Chakraborti, A., Leyvraz, F., & Seligman, T. H. (2018). Identifying long-term precursors of financial market crashes using correlation patterns. New Journal of Physics, 20(10), 103041.

[4] Pharasi, H. K., Seligman, E., & Seligman, T. H. (2020). Market states: A new understanding. arXiv preprint arXiv:2003.07058.

[5] Anirban Chakraborti, Kiran Sharma, Hirdesh K Pharasi, K Shuvo Bakar, Sourish Das and Thomas H Seligman; Emerging spectra characterization of catastrophic instabilities in complex systems (2020) New J. Phys. 22 063043.

[6] Manuel Mijail Martínez Ramos; Caracterización estadística de mercados europeos, Master thesis UNAM (2018).

[7] Vyas, M., Guhr, T., & Seligman, T. H. (2018). Multivariate analysis of short time series in terms of ensembles of correlation matrices. Scientific reports, 8(1), 1-12.

Abstract: The structure function, also known as the spectral form factor, is fundamental to the bulk scaled state in RMT. In the case of the circular β ensemble, we discuss a functional equation satisfied by this quantity, which relates β to 4/β. Its small k expansion has coefficients which are polynomials in β/2. Exact results for special β allow these to be computed up to 9th order. Their zeros have been conjectured to all lie on the unit circle in the complex β/2 plane, and have an interlacing property. A recent study of the structure function for even β using linear differential equations of degree β +1 allows the 10th order polynomial to be computed, and also provides extra insight into the functional forms for the classical values β = 1, 2 and 4.

Abstract: We consider the hypercubic model originally introduced by Parisi as a model for an array of Josephson junctions. This is a model where the Hamiltonian is given by the discretized Laplacian on a d-dimensional hypercube with U(1) gauge fields on the links but with a magnetic flux of constant magnitude and random orientation through all faces. It can also be expressed in terms of a tensor product of d qubits. In addition to the bipartite chiral symmetry, this model also has a magnetic inversion symmetry. Its ground state is separated from the rest of the spectrum by a gap. As is the case for the SYK model, the spectral density of this model is given by the density function of the Q-Hermite polynomials. We analyze the spectral correlations of this model and find that the spectral form factor and the number variance are in the universality class of the Gaussian Unitary Ensemble, while the eigenvalues near zero are described by the chiral Gaussian Unitary Ensemble.

Abstract: In random matrix theory, there has been great interest in establishing exact limit laws of statistical quantities and their corresponding rates of convergence. This talk will begin by introducing the Cauchy ensemble and the bijectively related circular Jacobi ensemble. Our discussion will subsequently focus on the correlation functions near the spectrum singularity (hard edge) of the circular Jacobi ensemble, their large N asymptotics, and the finite N corrections to these asymptotics. We will consider two cases via distinct approaches: The first case taking β∈{1,2,4} and the second case regarding β an even positive integer. We will then observe that upon appropriate scaling, the leading correction term can be optimally tuned to be proportional to 1/N^2.

Abstract: By the Bohigas-Giannoni-Schmit conjecture (1984), the spectral statistics of quantum systems whose classical counterparts exhibit chaotic behavior are described by random matrix theory. An alternative characterization of eigenvalue fluctuations was suggested where a long sequence of eigenlevels has been interpreted as a discrete-time random process. It has been conjectured that the power spectrum of energy level fluctuations shows 1/ω noise in the chaotic case, whereas, when the classical analog is fully integrable, it shows 1/ω^2 behavior. In the first part of this talk, I will introduce the definition of the power spectrum and consider its analysis in the case of the Circular Unitary Ensemble. Our theory produces a parameter-free prediction for the power spectrum expressed in terms of a fifth Painlevé transcendent. In the second part, I will show numerical results which uses zeros of the Riemann Zeta function. I will present a fair evidence that a universal Painlevé V curve can be observed in its power spectrum.

Abstract: We consider a large non-Hermitian i.i.d. matrix X with real or complex entries and show that the linear statistics of the eigenvalues are asymptotically Gaussian for test function having 2+ε derivatives. Previously this result was known only for the Ginibre ensemble where explicit formulas for the correlation functions are available; our result holds for general distribution of the matrix entries. The proof relies on two main novel ingredients: (i) local law for product of resolvents of the Hermitisation of X at two different spectral parameters, (ii) coupling of several weakly dependent Dyson Brownian motions.

Abstract: Using the supersymmetric method in form of the superbosonization formula [Littelmann, Sommers, Zirnbauer (2008)], we derive an explicit expression for the 1-point function of the shifted Ginibre ensemble in both the real and complex symmetric class. Our result implies an optimal lower bound on the least singular value of the shifted Ginibre ensemble which improves the classical smoothing bound from [Sankar, Spielman, Teng (2006)] in the transitional edge regime. Finally, we demonstrate how the optimal lower bound, together with a long-time Green function comparison argument, implies edge universality for i.i.d. matrices.

Abstract: We consider a sub-class of probability measures within determinantal point processes called polynomial ensembles. Examples of such ensembles include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We analyze expectation values of characteristic polynomials to obtain determinantal formulas for quantities such as the correlation kernel. This leads to the notion of invertibility in polynomial ensembles, which can be used to derive determinantal formulas only depending on the number of characteristic polynomials. The correlation kernels for two models, closely related to applications in effective field theory, are derived via these formulas for finite N. We perform large N asymptotic analysis of the two kernels and obtain universality results in form of Bessel-type kernels.

Abstract: This talk will start with a relaxed introduction to the combinatorics of the Gaussian and Laguerre unitary ensembles’ spectral moments. In particular, ribbon graph and combinatorial map interpretations will be given, along with their connection to ramified coverings of the Riemann sphere. Then, we will briefly look at the spectral moments of a matrix ensemble that behaves like a fusion of the GUE and LUE. One of the motivations for studying this “fusion” is that it serves as a matrix realisation of the Hermite Muttalib-Borodin ensemble in the large separation regime.

Abstract: Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the Circular Unitary Ensemble and other Circular Ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulae, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble of random matrices, and other related unitary invariant matrix ensembles. This allows us to write down exact formulae in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms multivariate orthogonal polynomials. This is joint work with Bhargavi Jonnadula and Jonathan P. Keating.

Abstract: An invariant ensemble is a random matrix ensemble that is unchanged under an adjoint action of a group of invariance. In this talk we will focus on sums/products of such ensembles, where two general formulae for their eigenvalue PDFs are provided, connecting eigenvalue PDFs to additive/multiplicative weights. Essentially, a matrix sum/product corresponds to the convolution of their weights. These results generalise the idea of Pólya ensembles introduced by Föster, Kieburg and Kösters.

Abstract: We consider random density matrices distributed according to the Hilbert-Schmidt and Bures-Hall probability measures and discuss some of our results concerning the spectral densities and the associated average entropies. We also touch upon our very recent result concerning the mean-square Hilbert-Schmidt distance between two random density matrices. Finally, we talk about the realization of these concepts and results in a system of coupled kicked tops. 

Abstract: In this talk Dr. Gurau will discuss a simple generalization of the resolvent to real symmetric tensors which yields a spectral representation of a subclass of tensor invariants. He will then discuss the expected resolvent for a random tensor distributed on a Gaussian (generalizing the Gaussian Orthogonal Ensemble) and show that the spectral density (discontinuity of the resolvent at the cut) respects a universal law generalizing the Wigner semicircle law to higher order tensors. Finally, Dr. Gurau will discuss an application of this resolvent to the spiked tensor model. 

Abstract: We discuss some recent computations of the exact moments of entanglement entropies over the Hilbert-Schmidt measure and the Bures-Hall measure. The emphasis is on the results of the latter measure, which are applications of a work by Peter Forrester and Mario Kieburg. 

Abstract: I will first explain how global rigidity upper bounds for universal random matrix point processes can be derived from asymptotics for their exponential moments. Secondly, I will apply this method to the Airy and Bessel point processes, for which exponential moment asymptotics are well-known. In the third part, I will focus on product random matrix determinantal point processes and on processes arising in Muttalib-Borodin ensembles: for these processes, I will show how one can obtain exponential moment asymptotics, and I will derive global rigidity results.

The talk will be based on joint work with Christophe Charlier (KTH Stockholm).

Abstract: I will start with a brief introduction to quantum information theory, emphasizing the central role played by quantum states and quantum channels. I will then discuss natural probability distributions of states and channels, motivated by physics and quantum Shannon theory. Applications, such as the distinguishability of quantum states and the minimum output entropy of quantum channels will be presented if time permits. 

Abstract: In this talk, I will discuss complex eigenvalues of the product of two rectangular complex Ginibre matrices that are correlated through a non-Hermiticity parameter. In the first half, I will present the limiting spectral distribution of the model, which interpolates between classical results for random matrices on the global scale, the circular law, and the Marchenko-Pastur distribution. In the second half, I will explain the microscopic behaviours of the model, which includes the limiting local correlation kernel at multi-criticality, where the interior of the spectrum splits into two connected components. The global statistics follows from the solution of certain equilibrium measure problem and concentration for the 2D Coulomb gases on Frostman’s equilibrium measure, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane. This is based on joint work with Gernot Akemann and Nam-Gyu Kang.

Abstract: In my recent works on the hard edge of random matrix models, I found that using the expansion of ratio of gamma functions to deal with the asymptotics is efficient. Along with the leading term, the correction term and optimal convergence rate can also be obtained. This talk is an illustrative talk to show the method. Many examples including Laguerre polynomials, Jacobi polynomials and the kernel of Cauchy two-matrix model (Meijer G-functions/kernels) are demonstrated.

Abstract: If the matrix X has i.i.d. real Gaussian entries, X^TX represents the Wishart-Laguerre ensemble. Inserting the elementary antisymmetric matrix J to form X^T JX results in an ensemble whose eigenvalue distribution is akin to the Laguerre Muttalib-Borodin ensemble. This allows us to make a curious connection to the singular values of the product of two independent matrices with i.i.d. complex Gaussian entries. I will go through this connection and highlight why it is curious from the loop-equation viewpoint.