  In this paper, we consider a sequence of selfadjoint matrices $A_n$ having a
limiting spectral distribution as $n\to \infty$, and we consider a sequence of
full flags $\{0\le p_1^n\le\ldots\le p_i^n\le\ldots\le 1_n\}$ chosen at random
according to the uniform measure on full flag manifolds. We are interested in
the behaviour of the extremal eigenvalues of $p_i^nA_np_i^n$. This problem is
known to be equivalent to the study of uniform probability measures on
Gelfand-Tsetlin polytopes. Our main results consist in explicit uniform
estimates for extremal eigenvalues, and the fact that an outlier behavior has
an exponentially small probability. This problem is of intrinsic interest in
random matrix theory, but it has also a strong motivation and some applications
in quantum information, which we discuss. The proofs rely on a reinterpretation
of the problem with the help of determinantal point processes and the
techniques are based on steepest descent analysis.
