T-4

Goal of these problems:

Key concepts:

RSB and the free-entry landscape

We have seen an example of mean-field model, the spherical p-spin, in which the low-T phase is glassy, described by a 1-RSB ansatz of the overlap matrix. The thermodynamics in the glassy phase is described by three quantities: the typical overlap $q_{1}^{SP}$ between configurations belonging to the same pure state, the typical overlap $q_{0}^{SP}$ between configurations belonging to different pure states, and the probability $(1-m^{SP})$ that two configurations extracted at equilibrium belong to the same state. It can be shown that the low-T, frozen phase of the REM is also described by this 1-RSB ansatz with $q_{0}^{SP}=0,q_{1}^{SP}=1$ and $m^{SP}=T/T_{c}$.