abstract

Recent work of Belinschi, Mai and Speicher resulted in a general algorithm to calculate the distribution of any self-adjoint polynomial in free variables. Since many classes of independent random matrices become asymptotically free if the size of the matrices goes to infinity, this algorithm allows then also the calculation of the asymptotic eigenvalue distribution of polynomials in such independent random matrices. We will recall the main ideas of this approach and then also present its extension to the case of the Brown measure of non-self-adjoint polynomials.

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Correlations play an important rôle when estimating risk in the financial markets. This is so from a systemic viewpoint when trying to assess the stability of the markets, but also from a practical one when, e.g. , optimizing portfolios. The non-stationarity of the correlations in time poses challenges for the modelling usually not encountered in the more traditional systems of statistical physics. Three recent results are presented and discussed. First it is shown how severely the exclusive look on the correlations can lead to misjudgements of the mutual dependencies. Second, the identification of distinct market states is reported and, third, generic features of return distributions are shown to be well captured by a random matrix model.

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In this article, we show the existence of limiting spectral distribution of a symmetric random matrix whose entries come from a stationary Gaussian process with covariances satisfying a summability condition. We provide an explicit description of the moments of the limiting measure. We also show that in some special cases the Gaussian assumption can be relaxed. The description of the limiting measure can also be made via its Stieltjes transform which is characterized as the solution of a functional equation. In two special cases, we get a description of the limiting measure — one as a free product convolution of two distributions, and the other one as a dilation of the Wigner semicircular law.

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We evaluate spectra of stochastic matrices defined on complex random graphs, with edges \((i,j)\) of a graph being given positive random weights \(W_{ij} \gt 0\) in such a fashion that column sums are normalized to one. The structure of the graphs and the distribution of the non-zero edge weights \(W_{ij}\) are largely arbitrary. We only require that the mean vertex degree remains finite in the thermodynamic limit, and that the \(W_{ij}\) satisfy a detailed balance condition. The main motivation for this work derives from the fact that knowing the spectra of stochastic matrices is tantamount to knowing the complete spectrum of relaxation times of stochastic processes described by them. One of the interesting new phenomena uncovered by our study is the appearance of localization transitions and mobility edges in the spectra of stochastic matrices of the type investigated in the present study.

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Let \(H_{n}\) be the \(n\times n\) symmetric Hankel-type matrix whose \((i,j)^{\mathrm {th}}\) element on the \(k^{\mathrm {th}}\) anti-diagonal (where \(k=0\) denotes the main anti-diagonal) is defined as: \(H_{k, n}(i,j)=g_k(\frac {i-[\frac {n+k+1}{2}]}{n})\) if \(i+j=n+1+k\) and 0 otherwise. Under suitable symmetry and summability conditions on \(\{g_k\}\), we show that as \(n \to \infty \), the limiting spectral distribution of \(\{H_n\}\) exists and is given by \(\sum _{k=-\infty }^{\infty } g_k(U) a_k\), where \(U\) is uniformly distributed on \([-1/2, \ 1/2]\) and is tensor-independent of the non-commutating variables \(\{a_k\}\) which are certain symmetric pair-wise free but not completely free Bernoulli variables.

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We characterize the phenomenon of “crowding” near the largest eigenvalue \(\lambda _{\max }\) of random \(N \times N\) matrices belonging to the Gaussian \(\beta \)-ensemble of random matrix theory, including, in particular, the Gaussian orthogonal (\(\beta =1\)), unitary (\(\beta =2\)) and symplectic (\(\beta = 4\)) ensembles. We focus on two distinct quantities: (i) the density of states (DOS) near \(\lambda _{\max }\), \(\rho _{\rm DOS}(r,N)\), which is the average density of eigenvalues located at a distance \(r\) from \(\lambda _{\max }\) (or the density of eigenvalues seen from \(\lambda _{\max }\)) and (ii) the probability density function of the gap between the first two largest eigenvalues, \(p_{\rm GAP}(r,N)\). Using heuristic arguments as well as numerical simulations, we generalize our recent exact analytical study of the Hermitian case (corresponding to \(\beta = 2\)). We also discuss some applications of these two quantities to statistical physics models.

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We consider the singular value statistics of products of independent random matrices. In particular, we compute the corresponding averages of products of characteristic polynomials. To this aim, we apply the projection formula recently introduced for chiral random matrix ensembles which serves as a shortcut of the supersymmetry method. The projection formula enables us to study the local statistics where free probability currently fails. To illustrate the projection formula and underlining the power of our approach, we calculate the hard edge scaling limit of the Meijer G-ensembles comprising the Wishart–Laguerre (chiral Gaussian), the Jacobi (truncated orthogonal, unitary or unitray symplectic) and the Cauchy–Lorentz (heavy tail) random matrix ensembles. All calculations are done for real, complex, and quaternion matrices in a unifying way. In the case of real and quaternion matrices, the results are completely new and hint to the universality of the hard edge scaling limit for a product of these matrices, too. Moreover, we identify the non-linear \(\sigma \)-models to the local statistics of product matrices at the hard edge.

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Our results concern quantum chaos of the vacuum Bianchi IX model. We apply the equilateral triangle potential well approximation to the potential of the Bianchi IX model to solve the eigenvalue problem for the physical Hamiltonian. Such approximation is well satisfied in vicinity of the cosmic singularity. Level spacing distribution of the eigenvalues is studied with and without applying the unfolding procedure. In both cases, the obtained distributions are qualitatively described by Brody’s distribution with the parameter \(\beta \approx 0.3\), revealing some sort of the level repulsion. The observed repulsion may reflect chaotic nature of the classical dynamics of the Bianchi IX universe. However, full understanding of this effects will require examination of the Bianchi IX model with the exact potential.

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We consider the products of \(m\ge 2\) independent large real random matrices with independent tuples \((X_{jk}^{(q)},X_{kj}^{(q)})\), \(1\le j \lt k\le n\) of entries. The entries \(X_{jk}^{(q)},X_{kj}^{(q)}\) are standardized and correlated with correlation coefficient \(\rho =\boldsymbol {E} [X_{jk}^{(q)}X_{kj}^{(q)}]\). The limit distribution of the empirical spectral distribution of the eigenvalues of such products does not depend on \(\rho \) and is equal to the distribution of the \(m^{\rm th}\) power of a uniformly distributed random variable on the unit disc.

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In this review, we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from ensembles of independent real, complex, or quaternionic Ginibre matrices, or truncated unitary matrices. Additional mixing within one ensemble between matrices and their inverses is also covered. Exact determinantal and Pfaffian expressions are given in terms of the respective kernels of orthogonal polynomials or functions. Here, we list all known cases and some straightforward generalisations. The asymptotic results for large matrix size include new microscopic universality classes at the origin and a generalisation of weak non-unitarity close to the unit circle. So far, in all other parts of the spectrum, the known standard universality classes have been identified. In the limit of infinite products, the Lyapunov and stability exponents share the same normal distribution. To leading order, they both follow a permanental point processes. Our focus is on presenting recent developments in this rapidly evolving area of research.

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Following Dyson, we treat the eigenvalues of a random matrix as a system of particles undergoing random walks. The dynamics of large matrices is then well described by fluid dynamical equations. In particular, the inviscid Burgers’ equation is ubiquitous and controls the behavior of the spectral density of large matrices. The solutions to this equation exhibit shocks that we interpret as the edges of the spectrum of eigenvalues. Going beyond the large \(N\) limit, we show that the average characteristic polynomial (or the average of the inverse characteristic polynomial) obeys equations that are equivalent to a viscid Burgers’ equation, or equivalently a diffusion equation, with \(1/N\) playing the role of the viscosity and encoding the entire finite \(N\) effects. This approach allows us to recover in an elementary way many results concerning the universal behavior of random matrix theories and to look at QCD spectral features from a new perspective.

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We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with the Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial differential, diffusion-like equations. These equations are valid for matrices of arbitrary size and for any initial condition assigned to the process. The solutions have compact integral representation that allows for a simple study of their asymptotic behavior, uncovering the Airy and Pearcey functions.

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We review some aspects of recent work concerning double scaling limits of singularly perturbed Hermitian random matrix models and their connection to Painlevé equations. We present new results showing how a Painlevé III hierarchy recently proposed by the author can be connected to the Lenard recursion formula used to construct the Painlevé I and II hierarchies.

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It has been shown by Voiculescu that important classes of square independent random matrices are asymptotically free, where freeness is a noncommutative analog of classical independence. Recently, we introduced the concept of matricial freeness, which is similar to freeness in free probability, but it also has some matricial features. Using this new concept of noncommutative independence, we described the asymptotics of blocks and symmetric blocks of certain classes of independent random matrices. In this paper, we present the main results obtained in this framework, concentrating on the ensembles of blocks of Gaussian random matrices.

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After reviewing the theory of trianguar (causal) and rectangular quantum stochastic double product integrals, we consider examples when these consist of unitary operators. We find an explicit form for all such rectangular product integrals which can be described as second quantizations. Causal products are proposed as paradigm limits of large random matrices in which the randomness is explicitly quantum or noncommutative in character.

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In this paper, we study a one-parameter deformation of the \(q\)-Laguerre weight function. An investigation is made on the polynomials orthogonal with respect to such a weight. With the aid of the two compatibility conditions, previously obtained in Y. Chen, M.E.H. Ismail, J. Math. Anal. Appl. 345, 1 (2008), and the \(q\)-analog of a sum rule obtained in this paper, we derive expressions for the recurrence coefficients in terms of certain auxiliary quantities, and show that these quantities satisfy a pair of first order non-linear difference equations. This manuscript is a shorter version of the full paper [Indagat. Math., to appear] and has been modified to be suitable for the proceedings of Matrix 2014.

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We review some results about roles of bistochastic matrices from a point of view of entropy and the notion of the relative position between subalgebras of matrix algebras.

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We show that microwave networks simulating quantum graphs are very useful in an experimental investigation of isoscattering phenomena in a broad frequency range from 0.01 to 5 GHz.