scholarly journals Large deviations for the largest eigenvalue of the sum of two random matrices

2020 ◽  
Vol 25 (0) ◽  
Author(s):  
Alice Guionnet ◽  
Mylène Maïda
Keyword(s):  
Large Deviations ◽  
Random Matrices ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

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 Large deviations for the largest eigenvalue of Rademacher matrices

2020 ◽  
Vol 48 (3) ◽  
pp. 1436-1465
Author(s):  
Alice Guionnet ◽  
Jonathan Husson
Keyword(s):  
Large Deviations ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue
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