Global marginality and stability away from criticality

Determine, for the Sherrington–Kirkpatrick model away from criticality and more generally for small constraints in the local regime, whether every positive contact of the constrained Parisi minimizer is marginal and whether its quantile function differs from that of the unconstrained Parisi measure by O(b8^5) above the first positive contact.

Background

The paper establishes marginality of the first positive contact for sufficiently small constraints, but its local plateau calculus controls contacts only in a fixed neighborhood of zero. It does not establish the behavior of all positive contacts farther from zero.

Near criticality, additional global control proves that positive contacts form an interval in the marginal regime and that the constrained and equilibrium quantile functions agree up to O(b85) above the first contact. The paper explicitly leaves the corresponding questions unresolved for the Sherrington–Kirkpatrick model away from criticality.

References

It leaves open whether, for small $\theta$, every positive contact of $\zeta_\theta$ is marginal, as the first one is, and whether the quantile functions of $\zeta_\theta$ and $$ differ by $O(\theta5)$ above the first contact, as Parisi and Rizzo predict at every temperature below the critical one Section~IV.

— Upper deviations of the Sherrington-Kirkpatrick free energy: the fifth-order law and the onset of constrained Parisi measures  (2610.06480 - Pei, 5 Oct 2026) in Section "Earlier work and open questions", subsection "Inputs and open questions", paragraph "Away from zero"