Patterned sparse random matrices: A moment approach

2017 ◽  
Vol 06 (03) ◽  
pp. 1750011
Author(s):  
Debapratim Banerjee ◽  
Arup Bose
Keyword(s):  
Random Matrices ◽  
Limit Distribution ◽  
Spectral Distribution ◽  
Explicit Description ◽  
Circulant Matrices ◽  
Hankel Matrices ◽  
Largest Eigenvalue ◽  
Empirical Spectral Distribution ◽  
Moment Approach ◽  
The Largest Eigenvalue

We consider four specific [Formula: see text] sparse patterned random matrices, namely the Symmetric Circulant, Reverse Circulant, Toeplitz and the Hankel matrices. The entries are assumed to be Bernoulli with success probability [Formula: see text] such that [Formula: see text] with [Formula: see text]. We use the moment approach to show that the expected empirical spectral distribution (EESD) converges weakly for all these sparse matrices. Unlike the Sparse Wigner matrices, here the random empirical spectral distribution (ESD) converges weakly to a random distribution. This weak convergence is only in the distribution sense. We give explicit description of the random limits of the ESD for Reverse Circulant and Circulant matrices. As in the non-sparse case, explicit description of the limits appears to be difficult to obtain in the Toeplitz and Hankel cases. We provide some properties of these limits. We then study the behavior of the largest eigenvalue of these matrices. We prove that for the Reverse Circulant and Symmetric Circulant matrices the limit distribution of the largest eigenvalue is a multiple of the Poisson. For Toeplitz and Hankel matrices we show that the non-degenerate limit distribution exists, but again it does not seem to be easy to obtain any explicit description.

scholarly journals PROBABILITY DENSITIES AND DISTRIBUTIONS FOR SPIKED AND GENERAL VARIANCE WISHART β-ENSEMBLES

2013 ◽  
Vol 02 (04) ◽  
pp. 1350011 ◽  
Author(s):  
PETER J. FORRESTER
Keyword(s):  
Random Matrices ◽  
Exact Form ◽  
Recursive Structure ◽  
Simple Derivation ◽  
Largest Eigenvalue ◽  
Wishart Matrices ◽  
Real Quaternion ◽  
Parameter Dependent ◽  
Dependent State ◽  
The Largest Eigenvalue

A Wishart matrix is said to be spiked when the underlying covariance matrix has a single eigenvalue b different from unity. As b increases through b = 2, a gap forms from the largest eigenvalue to the rest of the spectrum, and with b - 2 of order N-1/3 the scaled largest eigenvalues form a well-defined parameter dependent state. Recent works by Bloemendal and Virág [Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825], and Mo [Rank I real Wishart spiked model, Comm. Pure Appl. Math.65 (2012) 1528–1638], have quantified this parameter dependent state for real Wishart matrices from different viewpoints, and the former authors have done similarly for the spiked Wishart β-ensemble. The latter is defined in terms of certain random bidiagonal matrices. We use a recursive structure to give an alternative construction of the spiked and more generally the general variance Wishart β-ensemble, and we give the exact form of the joint eigenvalue PDF for the two matrices in the recurrence. In the case of real quaternion Wishart matrices (β = 4) the latter is recognized as having appeared in earlier studies on symmetrized last passage percolation, allowing the exact form of the scaled distribution of the largest eigenvalue to be given. This extends and simplifies earlier work of Wang, and is an alternative derivation to a result in [A. Bloemendal and B. Virág, Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825]. We also use the construction of the spiked Wishart β-ensemble from [A. Bloemendal and B. Virág, Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825] to give a simple derivation of the explicit form of the eigenvalue PDF.

scholarly journals Circular law for random matrices with unconditional log-concave distribution

2015 ◽  
Vol 17 (04) ◽  
pp. 1550020 ◽  
Author(s):  
Radosław Adamczak ◽  
Djalil Chafaï
Keyword(s):  
Random Matrices ◽  
Spectral Distribution ◽  
Real Matrix ◽  
Random Real ◽  
Empirical Spectral Distribution ◽  
Circular Law ◽  
Ginibre Ensemble ◽  
Mean And Variance ◽  
Uniform Law ◽  
Special Case

We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.

scholarly journals Limiting spectral distribution of a class of Hankel type random matrices

2015 ◽  
Vol 04 (03) ◽  
pp. 1550010
Author(s):  
Anirban Basak ◽  
Arup Bose ◽  
Soumendu Sundar Mukherjee
Keyword(s):  
Random Matrices ◽  
Spectral Distribution ◽  
Rayleigh Distribution ◽  
Circulant Matrices ◽  
Limiting Spectral Distribution ◽  
Atomic Distribution

We consider an indexed class of real symmetric random matrices which generalize the symmetric Hankel and Reverse Circulant matrices. We show that the limiting spectral distribution of these matrices exists almost surely and the limit is continuous in the index. We also study other properties of the limit and, in particular, explicitly characterize it for a certain subclass of matrices as a mixture of the atomic distribution at zero and the symmetrized Rayleigh distribution.

Some patterned matrices with independent entries

2020 ◽  
pp. 2150030
Author(s):  
Arup Bose ◽  
Koushik Saha ◽  
Priyanka Sen
Keyword(s):  
Random Matrices ◽  
Spectral Distribution ◽  
Finite Variance ◽  
Weighted Sum ◽  
Hankel Matrices ◽  
Special Cases ◽  
Common Thread ◽  
Different Types ◽  
Patterned Matrices ◽  
Variance Profile

Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the [Formula: see text]th moment of the limit equals a weighted sum over different types of pair-partitions of the set [Formula: see text] and are universal. Some results are also known for the sparse case. In this paper, we generalize these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.

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