Ultrametricity in an Inhomogeneous Simplest Spin Glass Model

In order to study the limits of validity of ultrametricity we imagine a system composed of two SK models weakly coupled among themselves. For this case one assumes, as usual, that each one of them separately obeys ultrametricity while it is an open question whether the relevant configurations of the total system intended as a whole can be so classified. For zero coupling the andwer is trivially negative. For arbitrarily small interaction energy, provided it is extensive, an analysis of the associated generalized random energy models suggests that ultrametricity is recovered.