  A random overlap structure (ROSt) is a measure on pairs (X,Q) where X is a
locally finite sequence in the real line with a maximum and Q a positive
semidefinite matrix of overlaps intrinsic to the particles X. Such a measure is
said to be quasi-stationary provided that the joint law of the gaps of X and
overlaps Q is stable under a stochastic evolution driven by a Gaussian sequence
with covariance Q. Aizenman et al. have shown that quasi-stationary ROSts serve
as an important computational tool in the study of the Sherrington-Kirkpatrick
(SK) spin-glass model from the perspective of cavity dynamics and the related
ROSt variational principle for its free energy. In this framework, the Parisi
solution is reflected in the ansatz that the overlap matrix exhibit a certain
hierarchical structure. Aizenman et al. have posed the question of whether the
ansatz could be explained by showing that the only ROSts that are
quasi-stationary in a robust sense are given by a special class of hierarchical
ROSts known as both the Ruelle Probability Cascades as well the GREM. Arguin
and Aizenman have given an affirmative answer in the special case that the set
of values S_Q taken on by the entries of Q is finite. We prove that this result
holds even when |S_Q| is infinite provided that Q satisfies the technical
condition that the closure of S_Q has no limit points from below. This is
relevant to the understanding of the ground states of the SK model, as they
satisfy |S_Q| = infinity.
