Correlation inequalities for higher-replica spin models and off-diagonal observables

Derive FKG- and GKS-type correlation inequalities for the higher-replica permutation-valued spin models associated with shallow random quantum circuits, and use them to prove that the Scrooge-ensemble approximation bounds extend from the rigorously treated two-replica diagonal-observable case to arbitrary fixed replica indices and positive semidefinite off-diagonal observables.

Background

The analytical proof establishes a relative-error bound only for the simplest nontrivial replica setting, namely k=2, s=1, and t=0, where the statistical-mechanical model reduces to a ferromagnetic Ising model. The proof uses FKG and related correlation inequalities that are available for the Ising model.

For higher replica indices, the relevant model has permutation-valued spins and retains a ferromagnetic structure, but the required correlation inequalities are not known. Extending the argument to arbitrary positive semidefinite observables is also unresolved because the contribution from bad contour configurations does not have a definite sign. Proving suitable inequalities would make the heuristic higher-moment and off-diagonal extensions rigorous and would strengthen the central Scrooge conjecture.

References

In future work, it would be interesting to explore whether correlation inequalities of FKG- and GKS-type can be derived in the more general spin models for higher replica indices, which would be useful in making these generalized arguments rigorous.

— Conditional dependence and Scrooge ensembles in shallow random quantum circuits  (2608.12255 - Liu et al., 12 Aug 2026) in Supplementary Material, Section “Heuristic argument for higher replica indices and non-diagonal observables”