Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials

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 $$(X_{ij}, X_{ji})$$ ( X i j , X j i ) are mutually independent Gaussian vectors with $$\mathbb {E}\,X_{ij} = 0$$ E X i j = 0 , $$\mathbb {E}\,X^2_{ij} = 1$$ E X i j 2 = 1 and $$\mathbb {E}\,X_{ij} X_{ji} = \rho $$ E X i j X j i = ρ . We are interested in the asymptotic distribution of eigenvalues of the matrix $$W =\frac{1}{N^2} X^2 X^{*2}$$ W = 1 N 2 X 2 X ∗ 2 . We show that this distribution is determined by its moments, and we provide a recurrence relation for these moments. We prove that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B: $$\begin{aligned} c_{2n} = \sum _{k=0}^n {\left( {\begin{array}{c}n\\ k\end{array}}\right) }^2 \rho ^{2k}. \end{aligned}$$ c 2 n = ∑ k = 0 n n k 2 ρ 2 k . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
D.O.I.
10.1007/s10959-016-0685-5
Publisher site
See Article on Publisher Site

Abstract

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 $$(X_{ij}, X_{ji})$$ ( X i j , X j i ) are mutually independent Gaussian vectors with $$\mathbb {E}\,X_{ij} = 0$$ E X i j = 0 , $$\mathbb {E}\,X^2_{ij} = 1$$ E X i j 2 = 1 and $$\mathbb {E}\,X_{ij} X_{ji} = \rho $$ E X i j X j i = ρ . We are interested in the asymptotic distribution of eigenvalues of the matrix $$W =\frac{1}{N^2} X^2 X^{*2}$$ W = 1 N 2 X 2 X ∗ 2 . We show that this distribution is determined by its moments, and we provide a recurrence relation for these moments. We prove that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B: $$\begin{aligned} c_{2n} = \sum _{k=0}^n {\left( {\begin{array}{c}n\\ k\end{array}}\right) }^2 \rho ^{2k}. \end{aligned}$$ c 2 n = ∑ k = 0 n n k 2 ρ 2 k .

Journal

Journal of Theoretical ProbabilitySpringer Journals

Published: Apr 12, 2016

References

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