• Corpus ID: 119160069

Elliptic law for real random matrices

@article{Naumov2012EllipticLF,
  title={Elliptic law for real random matrices},
  author={Alexey Naumov},
  journal={arXiv: Probability},
  year={2012}
}
  • A. Naumov
  • Published 8 January 2012
  • Mathematics
  • arXiv: Probability
In this paper we consider ensemble of random matrices $\X_n$ with independent identically distributed vectors $(X_{ij}, X_{ji})_{i \neq j}$ of entries. Under assumption of finite fourth moment of matrix entries it is proved that empirical spectral distribution of eigenvalues converges in probability to a uniform distribution on the ellipse. The axis of the ellipse are determined by correlation between $X_{12}$ and $X_{21}$. This result is called Elliptic Law. Limit distribution doesn't depend… 

Figures from this paper

On a Generalization of the Elliptic Law for Random Matrices
We consider the products of $m\ge 2$ independent large real random matrices with independent vectors $(X_{jk}^{(q)},X_{kj}^{(q)})$ of entries. The entries $X_{jk}^{(q)},X_{kj}^{(q)}$ are correlated
Spectrum of Heavy-Tailed Elliptic Random Matrices
TLDR
A general bound is shown on the least singular value of elliptic random matrices under no moment assumptions and the convergence of the matrices to a random operator on the Poisson Weighted Infinite Tree converges, in probability, to a deterministic limit.
Asymptotic distribution of singular values for matrices in a spherical ensemble
AbstractWe consider the asymptotic behavior of the singular values of a so-called spherical ensemble of random matrices of large dimension. These are matrices of the form XY−1, where X and Y are
The Elliptic Law
We show that, under some general assumptions on the entries of a random complex $n \times n$ matrix $X_n$, the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges to the uniform law
Central Limit Theorem for Linear Eigenvalue Statistics of Elliptic Random Matrices
We consider a class of elliptic random matrices which generalize two classical ensembles from random matrix theory: Wigner matrices and random matrices with iid entries. In particular, we establish a
Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials
We consider Gaussian elliptic random matrices X of a size $$N \times N$$N×N with parameter $$\rho $$ρ, i.e., matrices whose pairs of entries $$(X_{ij}, X_{ji})$$(Xij,Xji) are mutually independent
ON ONE GENERALIZATION OF THE ELLIPTIC LAW FOR
We consider the products of m � 2 independent large real ran- dom matrices with independent vectors (X (q) jk ,X (q) kj ) of entries. The entries X (q) jk ,X (q) kj are correlated with � = EX (q) jk
Low rank perturbations of large elliptic random matrices
TLDR
A number of interesting results concerning elliptic random matrices whose entries have finite fourth moment are proved; these results include a bound on the least singular value and the asymptotic behavior of the spectral radius.
Circular law for random block band matrices with genuinely sublinear bandwidth
TLDR
The key technical result is a least singular value bound for shifted random band block matrices with genuinely sublinear bandwidth, which improves on a result of Cook in the band matrix setting.
From random matrices to long range dependence
TLDR
It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process, which helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.
...
...

References

SHOWING 1-10 OF 18 REFERENCES
The Elliptic Law
We show that, under some general assumptions on the entries of a random complex $n \times n$ matrix $X_n$, the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges to the uniform law
Random matrices: Universality of local eigenvalue statistics
In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the
The Strong Elliptic Law. Twenty years later. Part I
The structure of this survey is the following: at first we repeat in Part I the first 20 years old proof of the strong Elliptic law for random matrices Then in Part II we prove the strong Elliptic
Invertibility of symmetric random matrices
  • R. Vershynin
  • Mathematics, Computer Science
    Random Struct. Algorithms
  • 2014
TLDR
It is shown that H is singular with probability at most exp(−nc) , and ||H−1||=O(n) .
Around the circular law
These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the
Topics in Random Matrix Theory
The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field
COMPLEX HERMITE POLYNOMIALS: FROM THE SEMI-CIRCULAR LAW TO THE CIRCULAR LAW
We study asymptotics of orthogonal polynomial measures of the form |HN| 2 d∞ where HN are real or complex Hermite polynomials with re- spect to the Gaussian measure ∞. By means of dierential
Spectral Analysis of Large Dimensional Random Matrices
Wigner Matrices and Semicircular Law.- Sample Covariance Matrices and the Mar#x010D enko-Pastur Law.- Product of Two Random Matrices.- Limits of Extreme Eigenvalues.- Spectrum Separation.-
Bilinear and quadratic variants on the Littlewood-Offord problem
If f(x1, …, xn) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear
...
...