  This paper deals with two 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 get very large, with
horizontal and vertical dominos being equally likely. This extends naturally a
result of Johansson and Nordenstam in [17], 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 sharing 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 notice that all statements concerning these
coupled random matrices have a domino tiling counterpart.
