Rotational invariance in the near‑critical regime for FK‑percolation with 1 ≤ q ≤ 4

Establish that for two‑dimensional FK‑percolation on the square lattice Z^2 with cluster‑weight q in [1,4], the correlation length ξ_{p,q}(θ) and the point‑to‑hyperplane decay rate ζ_{p,q}(θ) become asymptotically isotropic as p approaches the critical value p_c(q) from below; specifically, prove that for any angles θ1, θ2 in [0,2π), the ratios ξ_{p,q}(θ1)/ξ_{p,q}(θ2) → 1 and ζ_{p,q}(θ1)/ζ_{p,q}(θ2) → 1 as p ↗ p_c(q).

Background

The paper proves that for q > 4 at criticality, the Wulff shape becomes round as q ↓ 4, implying isotropic correlation lengths in the limit. Motivated by near‑critical behavior, the authors formulate analogous expectations for fixed q ∈ [1,4] as p approaches p_c(q) from below.

They state precise expectations that the directional correlation length and point‑to‑hyperplane decay become rotationally invariant in this near‑critical limit, but they are unable to prove these statements with their current methods. The main technical obstruction identified is that the star–triangle transformations used in their proof are exact only at criticality (p = p_c), not in the near‑critical regime (p ≠ p_c).

References

In particular, we expect that, for any 1 \leq q \leq 4 and any two angles \theta_1,\theta_2, \begin{align} \frac{\xi_{p,q}(\theta_1) }{\xi_{p,q}(\theta_2)} \to 1 \quad \text{and} \quad \frac{\zeta_{p,q}(\theta_1) }{\zeta_{p,q}(\theta_2)} \to 1 \qquad \text{ as $p\nearrow p_c(q)$}. \label{eq:conjectures_rot_inv_nc} \end{align} The authors have not managed to adapt the strategy below to also prove eq:conjectures_rot_inv_nc and we believe a key ingredient is missing.

— The Wulff crystal of self-dual FK-percolation becomes round when approaching criticality  (2603.16318 - Manolescu et al., 17 Mar 2026) in Subsection “Near-critical FK-percolation with q ≤ 4” (Section 1)