scholarly journals The limiting spectral distribution of the product of the Wigner matrix and a nonnegative definite matrix

2010 ◽  
Vol 101 (9) ◽  
pp. 1927-1949 ◽  
Author(s):  
Z.D. Bai ◽  
L.X. Zhang
Keyword(s):  
Spectral Distribution ◽  
Wigner Matrix ◽  
Limiting Spectral Distribution ◽  
Nonnegative Definite

scholarly journals SPECTRAL PROPERTIES OF RANDOM TRIANGULAR MATRICES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250003 ◽  
Author(s):  
RIDDHIPRATIM BASU ◽  
ARUP BOSE ◽  
SHIRSHENDU GANGULY ◽  
RAJAT SUBHRA HAZRA
Keyword(s):  
Spectral Distribution ◽  
Spectral Properties ◽  
Triangular Matrices ◽  
Wigner Matrix ◽  
Limiting Spectral Distribution ◽  
Convergence Of Sequences ◽  
Explicit Formulae ◽  
Patterned Matrices

We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.

scholarly journals Brown measure and asymptotic freeness of elliptic and related matrices

2019 ◽  
Vol 08 (02) ◽  
pp. 1950007
Author(s):  
Kartick Adhikari ◽  
Arup Bose
Keyword(s):  
Spectral Distribution ◽  
Limiting Spectral Distribution

We show that independent elliptic matrices converge to freely independent elliptic elements. Moreover, the elliptic matrices are asymptotically free with deterministic matrices under appropriate conditions. We compute the Brown measure of the product of elliptic elements. It turns out that this Brown measure is same as the limiting spectral distribution.

scholarly journals On the spectral distribution of large weighted random regular graphs

2014 ◽  
Vol 03 (04) ◽  
pp. 1450015 ◽  
Author(s):  
Leo Goldmakher ◽  
Cap Khoury ◽  
Steven J. Miller ◽  
Kesinee Ninsuwan
Keyword(s):  
Spectral Distribution ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Spectral Measures ◽  
Limiting Spectral Distribution ◽  
Random Weights ◽  
Large Trees ◽  
The Difference ◽  
The Relationship ◽  
Limiting Spectral Measure

McKay proved the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞. Given a large d-regular graph we assign random weights, drawn from some distribution [Formula: see text], to its edges. We study the relationship between [Formula: see text] and the associated limiting spectral distribution obtained by averaging over the weighted graphs. We establish the existence of a unique "eigendistribution" (a weight distribution [Formula: see text] such that the associated limiting spectral distribution is a rescaling of [Formula: see text]). Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d2)). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.

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