Breaking replica symmetry at two different temperatures

I will provide a basic introduction to the idea an overlap distribution to describe phase transitions in disordered systems and for a simple problem show how this can be calculated using Parisi's replica symmetry breaking scheme. I will then summarise our current research on how this can be extended to the overlap between 2 different temperatures. This research is in collaboration with Bernard Derrida.

The agenda.

1. Introduction to overlap distribution to describe phase transitions - ferromagnet vs spin glass.

2. Why we would like to know the overlap distribution between two different temperatures - chaos or not?

3. Random energy model - the simplest non-trivial model.

4. Overlap distribution in the random energy model and 1 step replica symmetry breaking at a single temperature.

5. Generalisation of the replica symmetry breaking scheme to 2 temperatures.

The basic concepts of overlap distributions and replicas are discussed in Mézard, M., Parisi, G. and Virasoro, M., 1987. Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications