T-4

Goal of these problems:

Key concepts:

RSB and the free-energy 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 $m^{SP}=T/T_{c}$. Replicas are a way to explore the structure of the free-energy landscape.

The Sherrington-Kirkpatrick model introduced in Lecture 1 also has a low-T phase that is glassy. However, the structure of the free-energy landscape is more complicated the mutual overlaps between equilibrium states are organized in a complicated pattern. To understand it, let us consider a 2-RSB ansatz for the overlap matrix: