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This paper deals with two Gaussian unitary ensemble (GUE)-matrices, coupled together through some inequalities between the spectra of the first few (small) principal minors. The main results of the paper is to show that the spectra of the principal minors of these coupled matrices behave statistically as the domino tilings of finitely overlapping Aztec diamonds when their sizes become very large, with horizontal and vertical dominos being equally likely. This extends naturally a result of Johansson and Nordenstam [20], stating that the spectra of the principal minors of a GUE-matrix behave statistically as domino tilings of an Aztec diamond, near the middle of its edge. Given the spectra of the two coupled matrices, the joint spectra of the underlying principal minors of the two GUE-matrices are uniformly distributed in a certain "double cone". In particular, this leads to two GUE-matrices, whose spectra share the same real line, with one spectrum being completely to the left of the other spectrum; this gives a new and simple extension of GUE. Also note that all statements concerning these coupled random matrices have a domino tiling counterpart. [ABSTRACT FROM AUTHOR]

Classically, it is well known that a single weight on a real interval leads to orthogonal polynomials. In Generalized orthogonal polynomials, discrete KP and Riemann-Hilbertproblems, Comm. Math. Phys. 207 (1999), 589–620, we have shown that m-periodic sequences of weights lead to “moments,” polynomials dened by determinants of matrices involving these moments and 2m + 1-step relations between them, thus leading to 2m + 1-band matrices L. Given a Darboux transformations on L, which effect does it have on the m-periodic sequence of weights and on the associated polynomials? These questions will receive a precise answer in this paper. The methods are based on introducing time parameters in the weights, making the band matrix L evolve according to the so-called discrete KP hierarchy. Darboux transformations on that L translate into vertex operators acting on the τ-function. [ABSTRACT FROM PUBLISHER]