Mean-field spin glasses: beyond Parisi's formula?
Spin glasses are models of statistical mechanics encoding disordered
interactions between many simple units. One of the fundamental
quantities of interest is the free energy of the model, in the limit
when the number of units tends to infinity. For a restricted class of
models, this limit was predicted by Parisi, and later rigorously proved
by Guerra and Talagrand. I will first show how to rephrase this result
using an infinite-dimensional Hamilton-Jacobi equation. I will then
present partial results suggesting that this new point of view may allow
to understand limit free energies for a larger class of models, focusing
in particular on the case in which the units are organized over two
layers, and only interact across layers.