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.

Background

The paper reviews Talagrand’s rigorous proof of the Parisi formula for the free energy in the Sherrington–Kirkpatrick (SK) model, resolving what had been a central open problem. Despite this resolution, the authors emphasize that several other fundamental aspects predicted by Parisi’s theory remain unproven.

These predictions pertain to structural and probabilistic properties of the Gibbs measure and related quantities in mean-field spin glass models. The authors point readers to further references for an overview, indicating that the landscape of unresolved questions is broad and significant.

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.

— Talagrand's mathematical journey to the Abel Prize 2024 (2410.07945 - Guédon et al., 9 Oct 2024) in Section 3, Talagrand's Work — Spin glasses and the Parisi formula (after Theorem: The Parisi formula)