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Products of correlated symmetric matrices and \(q\)-Catalan numbers. (English) Zbl 1020.15025
Wigner’s result about the convergence of the spectrum of random large symmetric matrices is extended to some products of, not necessarily Gaussian, correlated symmetric matrices. It is proved that an analogue of Wigner’s result holds for such products of matrices for which the laws of entries are not required to be Gaussian or symmetric.
The \(N \rightarrow \infty\) limit of the spectral measure of the products of the so-called reduced Wigner matrices is analyzed. A new phenomenon due to the correlation structure arises, namely a phase transition for the limiting first moments. Every limiting moment is written as a weighted enumeration of involutions or of rooted planar trees. The generating functions considered are continued fractions and the first moments in the case of Markovian correlations are closely connected with the well known combinatorial objects, called \(q\)-Catalan numbers.
Some results of B. F. Logan jun., J. E. Mazo, A. M. Odlyzko and L. A. Shepp [Bell System Tech. J. 62, 299–306 (1983)] concerning generating functions are analyzed, and finally it is proved by elementary calculations that the critical correlation is related to the least positive zero of the generalized Rogers-Ramanujan continued fraction, i.e. \(q\)-hypergeometric function.

15B52 Random matrices (algebraic aspects)
05A30 \(q\)-calculus and related topics
15A18 Eigenvalues, singular values, and eigenvectors
60F05 Central limit and other weak theorems
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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