Abstract
Let $X_{1}, \ldots , X_{m_{N}}$ be independent random matrices of order $N$ drawn from the polynomial ensembles of derivative type. For any fixed $n$, we consider the limiting distribution of the $n$th largest modulus of the eigenvalues of $X = \prod _{k=1}^{m_{N}}X_{k}$ as $N \to \infty $ where $m_{N}/N$ converges to some constant $\tau \in [0, \infty )$. In particular, we find that the limiting distributions of spectral radii behave like that of products of independent complex Ginibre matrices.
Highlights
Introduction and main resultsRecently, Kieburg and Kösters [13] showed that there is a one-to-one correspondence among the probability density functions of the so-called statistical isotropic matrices, those of the square of their singular values and those of their eigenvalues
We have investigated the asymptotic behavior of the kth large modulus of the eigenvalues of products of mN independent complex random matrices drawn http://www.imstat.org/ecp/
It was shown that the limiting distribution of kth large modulus of the eigenvalues is the same as that of the single complex Ginibre matrix [16] for mN /N → 0, while for mN /N → τ ∈ (0, ∞) the results are generalization of the results for the spectral radii of products of independent complex Ginibre matrices [11]
Summary
Introduction and main resultsRecently, Kieburg and Kösters [13] showed that there is a one-to-one correspondence among the probability density functions of the so-called statistical isotropic matrices, those of the square of their singular values and those of their eigenvalues. For a positive integer N , let X be a N × N complex random matrix with probability density function P with respect to the Lebesgue measure on CN×N . Let X1, X2 be two independent random matrices drawn from the polynomial matrix ensembles of derivative type with weight function w1, w2 respectively.
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