Uniqueness of infinite-volume random cluster measures outside criticality

Prove that, for every admissible single-site measure on the real-valued spin system, the set of inverse temperatures at which the infinite-volume free and wired random cluster measures differ is contained in the singleton consisting of the critical inverse temperature, and prove that this set is empty if and only if the phase transition is continuous.

Background

The paper defines Xi as the set of inverse temperatures beta for which the infinite-volume free and wired random cluster measures, Psi_beta0 and Psi_beta1, are distinct. The authors prove that Xi is at most countable and note that Xi is contained in the critical and supercritical regime.

The conjecture proposes a complete description of possible nonuniqueness: it should occur only at the critical point, and it should occur precisely in the case of a continuous phase transition. The statement would extend known results for particular models, including the Ising, phi4, and two-dimensional Blume–Capel models.

References

We propose the following conjecture regarding the critical and supercritical regimes. For every admissible single-site measure $\rho$, one has $\Xi\subset {\beta_c}$. Moreover, $\Xi=\emptyset$ if and only if the phase transition is continuous, i.e.~$m*(\beta_c)=0$.

— Supercritical sharpness for the random cluster representation of real-valued spin models  (2608.18045 - Gunaratnam et al., 18 Aug 2026) in Conjecture 1, Section 1, subsection “Mixing and consequences for the spin model”

However, continuity of the phase transition remains open for the Ising and $\varphi4$ models for these graphs beyond the case of $\mathbb{Z}d$.

— Supercritical sharpness for the random cluster representation of real-valued spin models  (2608.18045 - Gunaratnam et al., 18 Aug 2026) in Section 1, subsection “Mixing and consequences for the spin model”