Non-coexistence of the two rotated percolation processes

Determine whether, for every potential P in the class E on a transitive amenable graph of polynomial growth, the infinite-volume zero-field Gibbs measure satisfies ⟨·⟩ᵦ⁰ = ½⟨·⟩ᵦ⁺ + ½⟨·⟩ᵦ⁻, equivalently whether the difference in equation (infvolcor) vanishes at every temperature.

Background

The paper derives an infinite-volume identity expressing the difference between the plus-state and zero-field two-point functions through a connection event involving the coupled percolation configurations ω⁺ and ω⁻. For the Ising and φ⁴ models, this difference is known to vanish on transitive amenable graphs of polynomial growth.

The authors propose extending the resulting decomposition of the zero-field Gibbs state to every potential in the class E. Such an extension would amount, through the displayed identity, to a non-coexistence result for the two negatively associated rotated percolation processes, but the paper does not prove it.

References

We expect this to hold more generally for any potential $P \in E$, which in view of eq:infvolcor would corresponds to a non-coexistence result for the two negatively associated percolation processes $+$ and $-$. So far we were not able to prove this.

eq:infvolcor:

$\left< _x _y \right>^{+} -\left< _x _y \right>^{0} = 2 ^{+ / 0}_\beta[ ^+_x ^-_y 1( x \overset{^+}{\longleftrightarrow} \infty , \, y \overset{^-}{\longleftrightarrow} \infty )]. $

— Double cluster swapping for spin models: Pfaffian relations and sharpness  (2609.01362 - Engelenburg et al., 1 Sep 2026) in Section 3, subsection “The representation in infinite-volume,” paragraph following equation (infvolcor)