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

Author

Listed:
  • Nikita Alexeev

    (St. Petersburg State University
    George Washington University)

  • Alexander Tikhomirov

    (Department of Mathematics, Komi Science Center of Ural Division of RAS
    Syktyvkar State University)

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 .

Suggested Citation

  • Nikita Alexeev & Alexander Tikhomirov, 2017. "Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1170-1190, September.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:3:d:10.1007_s10959-016-0685-5
    DOI: 10.1007/s10959-016-0685-5
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