For spherical Spin Glasses at low temperature, the Gibbs measure can be “chaotic", i.e. depend very strongly on temperature.
In a recent joint work with E. Subag and O. Zeitouni, we give a very detailed description of the Gibbs measure for mixed spherical Spin Glasses at low temperature, in the so-called 1RSB phase (for one step Replica-Symmetry-Breaking phase). This detailed geometric description of the Gibbs measure implies the existence of chaos, and much more. This fact has to be contrasted with the case of pure spherical models, where chaos does not exist, as proved by E. Subag.