Remaining open predictions of Parisi’s spin glass theory
Ascertain the validity of the remaining fundamental predictions of Parisi’s theory of spin glasses, particularly in the context of mean-field models such as the Sherrington–Kirkpatrick model, by determining which predictions remain unproven and establishing rigorous proofs or counterexamples for them.
References
The Parisi conjecture, which is confirmed by Theorem \ref{thm:parisi formula}, was arguably the most famous open problem in the theory of spin glasses, but there are still quite a number of fundamental predictions of the Parisi theory that remain open problems to the present day and we refer the reader to, e.g., for a first overview.
Looking at the results for the 1RSB approximation of the two-group free energy and complexity, we can reasonably conjecture that the limit in Eq.~(\ref{eq:interpolation}) is exact: in the full-RSB two-group theory the complexity of marginal states vanishes exactly at the equilibrium free energy, i.e., the lowest-lying marginal TAP states are the equilibrium states of the Parisi solution, and there is no interval of free energies above $f_0{\rm eq}$ devoid of states.
What is the quenched expression of $x_P$ and whether it stays strictly positive along the quenched branch below $f{*}$ has not been determined yet.