Existence and selection of a closed two-step aging hierarchy

Establish whether the marginal replica-symmetry-breaking hierarchy for weakly coupled spherical p-spin systems closes at two steps or extends to further steps, and determine the associated solution-selection structure.

Background

To describe weak coupling, the paper introduces a marginal two-step replica-symmetry-breaking construction that can represent the distinct aging scales of the two subsystems. This construction improves the comparison with the dynamical data at a representative weak-coupling point, but its continuation behaves differently for different disorder correlations: in one case it connects to a marginal one-step solution, while in another it terminates while remaining two-step. The authors therefore leave unresolved whether two steps suffice globally or whether additional levels of replica-symmetry breaking are required.