Strict monotonicity/connected support of the Parisi measure in the Sherrington–Kirkpatrick model

Establish whether, for the Sherrington–Kirkpatrick (p=2) Ising spin glass model, the Parisi measure μ_{2,OPT} has a cumulative distribution function that is strictly increasing on its support, equivalently that the support of μ_{2,OPT} is a connected subset of [0,1], so that Montanari’s polynomial-time algorithm achieves near-optimality unconditionally.

Background

The article discusses recent algorithmic advances for optimizing the ground state of spin glasses via the Parisi variational framework. For p=2 (the Sherrington–Kirkpatrick model), Montanari constructed a polynomial-time algorithm that achieves any desired proximity to the Parisi-optimal value, but the guarantee is conditional on a structural property of the Parisi measure.

Specifically, the algorithm’s correctness relies on the conjecture that the cumulative distribution function associated with the Parisi measure is strictly increasing within its support, which is equivalent to the measure’s support being connected. Proving this would remove the remaining assumption and yield an unconditional near-optimal polynomial-time algorithm for the SK model, clarifying the precise relationship between ultrametric structure and algorithmic tractability.

References

Montanari has constructed a polynomial time algorithm which achieves any desired proximity $\eta_{p,\rm OPT}-\epsilon$ to optimality whp, assuming an unproven though widely believed conjecture that the cumulative distribution function associated with the Parisi measure $\mu_{p,\rm OPT}$ solving the variational problem (\ref{eq:Parisi-limit}) is strictly increasing within its support. Namely, the support of the measure induced by $\mu_{p,\rm OPT}$ is a connected subset of $[0,1]$.

— Turing in the shadows of Nobel and Abel: an algorithmic story behind two recent prizes (2501.15312 - Gamarnik, 25 Jan 2025) in Section “Ultrametricity guides algorithms for spin glasses”