Exponential mixing in the supercritical regime

Prove that, for every admissible single-site measure and every inverse temperature $\beta>\beta_c$, the unique infinite-volume random cluster measure satisfies an exponential mixing estimate between events supported in $\Lambda_n$ and events supported outside $\Lambda_{2n}$, uniformly in $n$, with an error bounded by $e^{-cn}$ for some $c>0$.

Background

The paper establishes almost-everywhere uniqueness of the infinite-volume random cluster measure but does not obtain quantitative mixing estimates. The proposed estimate concerns the sigma-algebras generated by the edge and absolute-value-field variables in an inner box and outside a box of twice the scale.

Such a bound would strengthen uniqueness to quantitative spatial decorrelation. The authors explain that, combined with their supercritical sharpness theorem, it would imply exponential decay of truncated correlations for the associated spin model. A corresponding non-perturbative result is known only for the Ising model using random-current methods.

References

A natural question is whether this can be upgraded to quantitative mixing estimates. In particular, we believe that the following exponential mixing property holds. For every $\beta>\beta_c$, there exists $c>0$ such that, for every $n\geq 1$, \begin{equation}\label{eq:exp_mixing} \sup_{\substack{A\in \sigma(\Lambda_n) \ B\in \sigma(\Lambda_{2n}c)} \big|\Psi_\beta[A\cap B] - \Psi_\beta[A]\Psi_\beta[B]\big| \leq e{-cn}, \end{equation} where $\sigma(\Lambda)$ denotes the $\sigma$-algebra generated by $(\omega_e){e\in \overline{E}(\Lambda)}$ and $(\mathsf{a}_x){x\in\Lambda}$, and where $\Psi_\beta$ denotes the (conjecturally) unique infinite volume measure at parameter $\beta$.

eq:exp_mixing:

sup⁡A∈σ(Λn)B∈σ(Λ2nc)∣Ψβ[A∩B]−Ψβ[A]Ψβ[B]∣≤e−cn,\sup_{\substack{A\in \sigma(\Lambda_n) \\ B\in \sigma(\Lambda_{2n}^c)}} \big|\Psi_\beta[A\cap B] - \Psi_\beta[A]\Psi_\beta[B]\big| \leq e^{-cn},

— Supercritical sharpness for the random cluster representation of real-valued spin models  (2608.18045 - Gunaratnam et al., 18 Aug 2026) in Conjecture 2, Section 1, subsection “Mixing and consequences for the spin model”

It remains to be checked whether one can make it work for this specific application.

— Cluster Representation of Renormalization Group Transformations and a Rigorous Proof for Convergence of the RG-Flow of the Ising Model to Trivial Fixed Points away from Criticality  (2608.18862 - Arz, 19 Aug 2026) in Section 5, subsection “Generalizations”