The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important difference is that the population covariance matrices, which are assumed to be non-random in the standard setting of statistics and random matrix theory, are now random, moreover, are certain functions of random data matrices. The problem has been considered in recent work [21] by using the techniques of free probability theory. Since, however, free probability theory deals with population matrices which are independent of the data matrices, its applicability in this case requires an additional justification. We present this justification by using a version of the standard techniques of random matrix theory under the assumption that the entries of data matrices are independent Gaussian random variables. In the subsequent paper [18] we extend our results to the case where the entries of data matrices are just independent identically distributed random variables with several finite moments. This, in particular, extends the property of the so-called macroscopic universality on the considered random matrices.
We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.
We study the distribution of singular values of product of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the population covariance matrices assumed to be non-random or random but independent of the random data matrix in statistics and random matrix theory are now certain functions of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [25, 13] by using the techniques of free probability…
We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance $N^{-3/4+β}$ for some positive $β>0$. We prove that the local eigenvalue statistics follows the universal Dyson sine kernel.
Let $D$ be a space of $2\times n$ matrices. Then the face of the cone of all completely positive maps from $M_2$ into $M_n$ given by $D$ is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in $D$ forms a subspace of $D$ together with zero and $D^\perp$ is spanned by rank one matrices.
We investigate the spectral fluctuation properties of constrained ensembles of random matrices (defined by the condition that a number N(Q) of matrix elements vanish identically; that condition is imposed in unitarily invariant form) in the limit of large matrix dimension. We show that as long as N(Q) is smaller than a critical value (at which the quadratic level repulsion of the Gaussian unitary ensemble of random matrices may be destroyed) all spectral fluctuation measures have the same form as for the Gaussian unitary ensemble.
Lie groups with two different root lengths allow two mixed sign homomorphisms on their corresponding Weyl groups, which in turn give rise to two families of hybrid Weyl group orbit functions and characters. In this paper we extend the ideas leading to the Gaussian cubature formulas for families of polynomials arising from the characters of irreducible representations of any simple Lie group, to new cubature formulas based on the corresponding hybrid characters. These formulas are new forms of Gaussian cubature in the short root length case and new forms of Radau cubature in the long root case.…
This paper is a continuation of our paper "Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices", J. Stat. Phys. (134), 147--159 (2009), in which we proved the Central Limit Theorem for the matrix elements of differential functions of the real symmetric random Gaussian matrices (GOE). Here we consider the real symmetric random Wigner matrices having independent (modulo symmetry conditions) but not necessarily Gaussian entries. We show that in this case the matrix elements of sufficiently smooth functions of these random matrices have in general another limiting law …
Products of random $2\times 2$ matrices exhibit Gaussian fluctuations around almost surely convergent Lyapunov exponents. In this paper, the distribution of the random matrices is supported by a small neighborhood of order $λ>0$ of the identity matrix. The Lyapunov exponent and the variance of the Gaussian fluctuations are calculated perturbatively in $λ$ and this requires a detailed analysis of the associated random dynamical system on the unit circle and its invariant measure. The result applies to anomalies and band edges of one-dimensional random Schrödinger operators.
This thesis reviews recent progress on products of random matrices from the perspective of exactly solved Gaussian random matrix models. We derive exact formulae for the correlation functions for the eigen- and singular values at arbitrary matrix dimension and for an arbitrary number of factors. These exact results are used to study asymptotic limits for the macroscopic densities and the microscopic correlations as either the matrix dimension or the number of factors tends to infinity.
Motivated by isotropic fully developed turbulence, we define a theory of symmetric matrix valued isotropic Gaussian multiplicative chaos. Our construction extends the scalar theory developed by J.P. Kahane in 1985.
For random matrix ensembles with non-gaussian matrix elements that may exhibit some correlations, it is shown that centered traces of polynomials in the matrix converge in distribution to a Gaussian process whose covariance matrix is diagonal in the basis of Chebyshev polynomials. The proof is combinatorial and adapts Wigner's argument showing the convergence of the density of states to the semicircle law.
We continue investigations of our previous papers, in which there were proved central limit theorems (CLT) for linear eigenvalue statistics Tr f(M_n) and there were found the limiting probability laws for the normalised matrix elements of differential functions f(M_n) of nxn real symmetric Wigner matrices M_n. Here we consider another spectral characteristic of Wigner matrices, namely, Tr f(M_n)A_n, where {A_n, n=1,2,...} is a certain sequence of non-random matrices. We show first that if M_n belongs to the Gaussian Orthogonal Ensemble, then Tr f(M_n)A_n satisfies CLT. Then we consider Wigner …
We study Yang-Baxter sigma models with deformed 4D Minkowski spacetimes arising from classical $r$-matrices associated with $κ$-deformations of the Poincaré algebra. These classical $κ$-Poincaré $r$-matrices describe three kinds of deformations: 1) the standard deformation, 2) the tachyonic deformation, and 3) the light-cone deformation. For each deformation, the metric and two-form $B$-field are computed from the associated $r$-matrix. The first two deformations, related to the modified classical Yang-Baxter equation, lead to T-duals of dS$_4$ and AdS$_4$\,, respectively. The third deformatio…
We provide a proof to a recent conjecture by Forrester [2014 J. Phys. A: Math. Theor. 47, 065202] regarding the algebraic and arithmetic structure of Meijer G-functions which appear in the expression for probability of all eigenvalues real for product of two real Gaussian matrices. In the process we come across several interesting identities involving Meijer G-functions.
We describe an elementary method to get non-asymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. As the basic example, we consider the GUE matrices. Immediate applications include GOE and the ensemble of Gaussian anti-symmetric Hermitian matrices. The estimates we derive give asymptotically exact expressions for the first terms of 1/N-expansions of the moments and covariance terms. We apply our method to the ensemble of Gaussian Hermitian random band matrices whose elements are zero outside of the band of width b. The estim…
We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices $H=W\tilde{H}W$, where $\tilde{H}$ is a random matrix from Gaussian unitary ensemble and $W$ is a deterministic diagonal matrix with positive entries. Using the supersymmetry approach we calculate analytically the moments and the distribution function of the eigenvectors components for a generic matrix $W$. We show that specific choices of $W$ can modify significantly the nature of the eigenvectors changing them from extended to critical to localized. Our analytical results are supported by numerical simulations…
The paper provides a new integral formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternion-valued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large and compare the Lyapunov exponents in models with a spike and no spikes.
We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that they both have power-law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions.
There is a natural action of the braid group on the symmetric matrices with units on the diagonal, appearing in various fields as Singularity Theory, Frobenius Manifolds or Isomonodromic deformations of certain classes of linear differential systems. The problem of finding the finite orbits of this action for the case of non-degenerate three-dimensional matrices was solved and it was conjectured that the finite orbits on any non-degenerate matrices correspond to finite Coxeter groups and conjugacy classes of quasicoxeter elements in them. In the present work it is considered another extremal c…
We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate eigenvalues, the real ones show characteristics of an intermediate incomplete spectrum, that is, of a so-called thinned ensemble. On the other hand, the complex ones show repulsion compatible with cubic-order repulsion of non normal matrices for the real matrices, but higher order repulsion for the complex and quaternion matrices.
In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of random matrices. This connection can be explained via point processes associated to the PNG model and the random matrices ensemble by an extension to the multilayer PNG and multi-matrix models, respectively. We also explain other models which are equivalent to the PNG model: directed polymers, the l…
The aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the six-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian Unitary Ensemble and the GUE-corners process.
We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M independent NxN Gaussian random matrices in the large-N limit is rotationally symmetric in the complex plane and is given by a simple expression rho(z,\bar{z}) = 1/(Mπσ^2} |z|^{-2+2/M} for |z|<σ, and is zero for |z|> σ. The parameter σcorresponds to the radius of the circular support and is related to the amplitude of the Gaussian fluctuations. This form of the eigenvalue density is highly universal. It is identical for products of Gaussian Hermitian, non-Hermitian, real or complex random matrices. It does not change even …
We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices, i.e. of the hermitian matrices $H_n$ with independent Gaussian entries such that $< H_{ij}H_{lk}>=δ_{ik}δ_{jl}J_{ij}$, where $J=(-W^2\triangle+1)^{-1}$. Assuming that $W^2=n^{1+θ}$, $0<θ<1$, we show that this asymptotic behavior (as $n\to\infty$) in the bulk of the spectrum coincides with those for the Gaussian Unitary Ensemble.
We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an $n \times n$ matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let $x_k$ denote eigenvalue number $k$. Under the condition that both $k$ and $n-k$ tend to infinity with $n$, we show that $x_k$ is normally distributed in the limit. We also consider the joint limit distribution of $m$ eigenvalues from the GOE or GSE with similar conditions on the indices. The result is an $m$-dimensional normal dist…
We present further examples of the correspondence between solutions of type IIB supergravity and classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). In the previous works, classical $r$-matrices have been composed of generators of only one of either $\mathfrak{so}(2,4)$ or $\mathfrak{so}(6)$. In this paper, we consider some examples of $r$-matrices with both of them. The $r$-matrices of this kind contain (generalized) Schrödinger spacetimes and gravity duals of dipole theories. It is known that the generalized Schrödinger spacetimes can also be obtained via a certain c…
Consider the product $X = X_{1}\cdots X_{m}$ of $m$ independent $n\times n$ iid random matrices. When $m$ is fixed and the dimension $n$ tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of $X$ for analytic test functions. We show that the limiting variance is universal in the sense that it does not depend on $m$ (the number of factor matrices) or on the distribution of the entries of the matrices. The main result generalizes and improves upon previous limit statements for the linear spectral statistics of a single iid matrix by Rider and Silverstein as we…
We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called $L^2$-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher-Hartwig singularities. Using Riemann-Hilbert methods, we prove a rather general Fisher-Hartwig formula for one-cut regular unitary invariant ensembles.
The exact value of the Lyapunov exponents for the random matrix product $P_N = A_N A_{N-1}...A_1$ with each $A_i = Σ^{1/2} G_i^{\rm c}$, where $Σ$ is a fixed $d \times d$ positive definite matrix and $G_i^{\rm c}$ a $d \times d$ complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.
We study the asymptotic behavior of the eigenvalues of Gaussian perturbations of large Hermitian random matrices for which the limiting eigenvalue density vanishes at a singular interior point or vanishes faster than a square root at a singular edge point. First, we show that the singular behavior propagates macroscopically for sufficiently small Gaussian perturbations, and we describe the macroscopic eigenvalue behavior for Gaussian perturbations of critical size. Secondly, for sufficiently small Gaussian perturbations of unitary invariant random matrices, we prove that the microscopic eigenv…
We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case we develop a new approach to obtain complex integral representations for densities of measures whose Stieltjes transforms satisfy algebraic equations of a certain type. In the special cases in which at most one factor of the product is a complex Gaussian we derive elementary expression…
We study operators obtained by coupling an $n \times n$ random matrix from one of the Gaussian ensembles to the discrete Laplacian. We find the joint distribution of the eigenvalues and resonances of such operators. This is one of the possible mathematical models for quantum scattering in a complex physical system with one semi-infinite lead attached.
For tri-diagonal matrices arising in the simplified Jaynes--Cummings model, we give an asymptotics of the eigenvalues, prove a trace formula and show that the Spectral Riemann Surface is irreducible.
Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly $k$ real eigenvalues as a determina…
We show that the global fluctuations of spectra of GOE and GUE matrices and their principal submatrices executing Dyson's Brownian motion are Gaussian in the limit of large matrix dimensions. For nested submatrices one obtains a limiting three-dimensional generalized Gaussian process; its restrictions to two-dimensional sections that are monotone in matrix sizes and time moments coincide with the two-dimensional Gaussian Free Field with zero boundary conditions. The proof is by moment convergence, and it extends to more general Wigner matrices and their stochastic evolution.
In this paper we study the distribution of level crossings for the spectra of linear families A+lambda B, where A and B are square matrices independently chosen from some given Gaussian ensemble and lambda is a complex-valued parameter. We formulate a number of theoretical and numerical results for the classical Gaussian ensembles and some generalisations. Besides, we present intriguing numerical information about the distribution of monodromy in case of linear families for the classical Gaussian ensembles of 3 * 3 matrices.
We prove that for Gaussian random normal matrices the correlation function has universal behavior. Using the technique of orthogonal polynomials and identities similar to the Christoffel-Darboux formula, we find that in the limit, as the dimension of the matrix tends to infinity, the density of eigenvalues converges to a constant inside of an ellipse and to zero outside. The convergence holds locally uniformly. At the boundary, in scaled coordinates holding the distance between eigenvalues constant, we show that the density is falling off to zero like the complementary error function. The conv…
Higher-order spacing ratios are investigated analytically using a Wigner-like surmise for Gaussian ensembles of random matrices. For $k$-th order spacing ratio $(r^{(k)}$, $k>1)$ the matrix of dimension $2k+1$ is considered. A universal scaling relation for this ratio, known from earlier numerical studies, is proved in the asymptotic limits of $r^{(k)}\rightarrow0$ and $r^{(k)}\rightarrow \infty$.
We consider four nontrivial ensembles involving Gaussian Wigner and Wishart matrices. These are relevant to problems ranging from multiantenna communication to random supergravity. We derive the matrix probability density, as well as the eigenvalue densities for these ensembles. In all cases the joint eigenvalue density exhibits a biorthogonal structure. A determinantal representation, based on a generalization of Andréief's integration formula, is used to compactly express the $r$-point correlation function of eigenvalues. This representation circumvents the complications encountered in the u…
We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dyson's Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies on cases where the initial density vanishes at an interior point of the support. We show that the time to reach universality becomes larger if the density vanishes faster or if the initial points …
We consider the mixed-moments $φ(\mathbf{X}^{ε_1},\ldots,\mathbf{X}^{ε_k})=\lim_{N\to\infty}N^{-1}\mathbb{E}\left[\mathrm{Tr}\left(\mathbf{X}^ε_1\cdots\mathbf{X}^{ε_k}\right)\right]$ of complex Gaussian Elliptic Matrices $\mathbf{X}$ (with correlation parameter $ρ$ between elements $\mathbf{X}_{ij}$ and $\mathbf{X}_{ji}^*$), where symbolically $ε_i\in\{1,\dagger\}$, and where the expectation $\mathbb{E}\left[\cdot\right]$ is taken over all matrices $\mathbf{X}$. We start by finding an explicit formula for $φ(\mathbf{X}^n,(\mathbf{X}^\dagger)^m)$, $n,m\in\mathbb{N}$, by using a mapping between …
We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N_{+}) of a random NxN matrix belonging to Gaussian orthogonal (β=1), unitary (β=2) or symplectic (β=4) ensembles. The distribution of the fraction of positive eigenvalues c=N_{+}/N scales, for large N, as Prob(c,N)\simeq\exp[-βN^2 Φ(c)] where the rate function Φ(c), symmetric around c=1/2 and universal (independent of $β$), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic sin…
