Marchenko Pastur Research Articles

Asymptotic Behavior of the Maximum and Minimum Singular Value of Random Vandermonde Matrices

This study examines various statistical distributions in connection with random Vandermonde matrices and their extension to $$d$$ -dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to be $$O(\log ^{1/2}{N^{d}})$$ and $$\Omega ((\log N^{d} /(\log \log N^d))^{1/2})$$ , respectively, where $$N$$ is the dimension of the matrix, generalizing the results in Tucci and Whiting (IEEE Trans Inf Theory 57(6):3938–3954, 2011). We further study the behavior of the minimum singular value of these random matrices. In particular, we prove that the minimum singular value is at most $$N\exp (-C\sqrt{N}))$$ with high probability where $$C$$ is a constant independent of $$N$$ . Furthermore, the value of the constant $$C$$ is determined explicitly. The main result is obtained in two different ways. One approach uses techniques from stochastic processes and in particular a construction related to the Brownian bridge. The other one is a more direct analytical approach involving combinatorics and complex analysis. As a consequence, we obtain a lower bound for the maximum absolute value of a random complex polynomial on the unit circle, which may be of independent mathematical interest. Lastly, for each sequence of positive integers $$\{k_p\}_{p=1}^{\infty }$$ we present a generalized version of the previously discussed matrices. The classical random Vandermonde matrix corresponds to the sequence $$k_{p}=p-1$$ . We find a combinatorial formula for their moments and show that the limit eigenvalue distribution converges to a probability measure supported on $$[0,\infty )$$ . Finally, we show that for the sequence $$k_p=2^{p}$$ the limit eigenvalue distribution is the famous Marchenko–Pastur distribution.

Convergence rates to the Marchenko–Pastur type distribution

S n = 1 n T n 1 / 2 X n X n ∗ T n 1 / 2 , where X n = ( x i j ) is a p × n matrix consisting of independent complex entries with mean zero and variance one, T n is a p × p nonrandom positive definite Hermitian matrix with spectral norm uniformly bounded in p . In this paper, if sup n sup i , j E ∣ x i j 8 ∣ < ∞ and y n = p / n < 1 uniformly as n → ∞ , we obtain that the rate of the expected empirical spectral distribution of S n converging to its limit spectral distribution is O ( n − 1 / 2 ) . Moreover, under the same assumption, we prove that for any η > 0 , the rates of the convergence of the empirical spectral distribution of S n in probability and the almost sure convergence are O ( n − 2 / 5 ) and O ( n − 2 / 5 + η ) respectively.

On a surprising relation between the Marchenko–Pastur law, rectangular and square free convolutions

Dans cet article, on prouve un résultat reliant les versions carré et rectangulaire de la R-transformée, qui a pour conséquence une relation surprenante entre les versions carré et rectangulaire de la convolution libre additive, impliquant la loi de Marchenko–Pastur. On donne des conséquences de ce résultat portant sur les matrices aléatoires, sur l’infinie divisibilité et sur l’arithmétique des versions carré des convolutions additives et multiplicatives.

On free and classical type G distributions

There is a one-to-one correspondence between classical one-dimensional infinitely divisible distributions and free infinitely divisible distributions. In this work we study the free infinitely divisible distributions corresponding to the one-dimensional type G distributions. A new characterization of classical type G distributions is given first and the class of type A classical infinitely divisible distributions is introduced. The corresponding free type A distributions are studied and the role of a special symmetric beta distribution is shown as a building block for free type A distributions. It is proved that this symmetric beta distribution is the free multiplicative convolution of an arcsine distribution with the Marchenko–Pastur distribution.

Random graph states, maximal flow and Fuss–Catalan distributions

For any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2m subsystems. Each edge of the graph represents a bipartite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated with a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze the statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko–Pastur) distribution. We derive a discrete family of generalized, Fuss–Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.

Circular Law for Noncentral Random Matrices

Let (X jk )j,k≥1 be an infinite array of i.i.d. complex random variables with mean 0 and variance 1. Let λ n,1,…,λ n,n be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leqslant j,k\leqslant n}$ . The strong circular law theorem states that, with probability one, the empirical spectral distribution $\frac{1}{n}(\delta _{\lambda _{n,1}}+\cdots+\delta _{\lambda _{n,n}})$ converges weakly as n→∞ to the uniform law over the unit disc {z∈ℂ,|z|≤1}. In this short paper, we provide an elementary argument that allows us to add a deterministic matrix M to (X jk )1≤ j,k ≤ n provided that Tr(MM *)=O(n 2) and rank(M)=O(n α ) with α<1. Conveniently, the argument is similar to the one used for the noncentral version of the Wigner and Marchenko–Pastur theorems.

The Rate of Convergence of Spectra of Sample Covariance Matrices

It is shown that the Kolmogorov distance between the spectral distribution function of a random covariance matrix ${\frac{1}{p}} {\bf X}\,{\bf X}^T$, where ${\bf X}$ is an $n\times p$ matrix with independent entries and the distribution function of the Marchenko–Pastur law is of order $O(n^{-1/2})$. The bounds hold uniformly for any p, including ${\frac{p}{n}}$ equal or close to 1.

Lyapunov exponents of free operators

Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede–Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko–Pastur law, and relate this example to C.M. Newman's “triangle” law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede–Kadison determinant and Voiculescu's S-transform.

Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

We develop a tool to approximate the entries of a large dimensional complex Jacobi ensemble with independent complex Gaussian random variables. Based on this and the author’s earlier work in this direction, we obtain the Tracy–Widom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko–Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.

Non-white Wishart ensembles

We consider non-white Wishart ensembles 1 p X Σ X * , where X is a p × N random matrix with i.i.d. complex standard Gaussian entries and Σ is a covariance matrix, with fixed eigenvalues, close to the identity matrix. We prove that the largest eigenvalue of such random matrix ensembles exhibits a universal behavior in the large-N limit, provided Σ is “close enough” to the identity matrix. If not, we identify the limiting distribution of the largest eigenvalues, focusing on the case where the largest eigenvalues almost surely exit the support of the limiting Marchenko–Pastur's distribution.

Classical and free infinitely divisible distributions and random matrices

We construct a random matrix model for the bijection Ψ between clas- sical and free infinitely divisible distributions: for every d≥1, we associate in a quite natural way to each *-infinitely divisible distribution μ a distribution ℙdμ on the space of d×d Hermitian matrices such that ℙdμ*ℙdν=ℙdμ*ν. The spectral distribution of a random matrix with distribution ℙdμ converges in probability to Ψ(μ) when d tends to +∞. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the Marchenko–Pastur distribution. In an analogous way, for every d≥1, we associate to each *-infinitely divisible distribution μ, a distribution $\mathbb{L}_{d}^{\mu}$ on the space of complex (non-Hermitian) d×d random matrices. If μ is symmetric, the symmetrization of the spectral distribution of |Md|, when Md is $\mathbb{L}_{d}^{\mu}$-distributed, converges in probability to Ψ(μ).

Elliptic Law

Elliptic Law

Computational Aspects of Three-Term Recurrence Relations

Computational Aspects of Three-Term Recurrence Relations