Asymptotic Pfaffian relations for the critical two-dimensional φ⁴ model

Establish whether Pfaffian relations hold asymptotically at the critical point for the finite-range φ⁴ model on the two-dimensional lattice ℤ².

Background

The paper studies Pfaffian relations, which express higher-order spin correlations in terms of two-point correlations. Such relations are known exactly for boundary spins of planar Ising models, and the paper proves a converse characterization for general classical spin systems.

The cited prior work formulated the unresolved question of whether these relations persist asymptotically at criticality for the two-dimensional φ⁴ model, a continuous-spin model expected to share important critical behavior with the Ising model.

References

In recent years new graphical proofs of this identity appeared using the double random current representation~\cites{LisT,ADTW}. Another possible approach is to directly use the exact solvability of the planar Ising model in the form of its dimer representation on the Fisher graph or the Kac--Ward solution. We also mention that Pfaffian relations have been shown to hold asymptotically at the critical point of finite-range Ising models on $Z2$ and it is conjectured that they also hold asymptotically for the critical $4$ model in the same setting Problem 2.

— Double cluster swapping for spin models: Pfaffian relations and sharpness  (2609.01362 - Engelenburg et al., 1 Sep 2026) in Section 2, subsection “Setup and result,” paragraph “Background”