Conformal convergence of FK-decorated planar maps to LQG and CLE

Prove that conformally embedded FK(q)-decorated planar maps with q in [0,4] converge, under suitable rescaling as the number of faces tends to infinity, to the corresponding gamma-Liouville quantum gravity surface jointly with an independent CLE_{16/gamma^2}, including the critical q=4 case with gamma=2 and CLE_4.

Background

The paper states the expected universality relation q = 2 + 2 cos(pi gamma2/2), with gamma in [sqrt(2),2]. Under this prediction, FK-decorated maps should converge after canonical conformal embedding and rescaling to gamma-Liouville quantum gravity decorated by an independent conformal loop ensemble. The authors formulate this as a conjectural scaling-limit statement; the paper proves only peanosphere convergence for the critical q=4 model, not the full conformal embedding convergence described here.

References

In particular, after conformally embedding the map into $\mathbb{C}$ {in a canonical way}, and scaling so that $N$ faces are mapped to the unit disc, the embedded metric measure space is conjectured to converge under suitable rescaling to a $\gamma$-LQG surface as $N\to \infty$. Moreover, the loops separating primal and dual FK clusters are expected to converge jointly with the surface to an independent conformal loop ensemble $\mathrm{CLE}_{16/\gamma2}$.

— Scaling limits of critical FK-decorated random planar maps with $q=4$  (2511.21480 - Silva et al., 26 Nov 2025) in Section 1, Introduction

Our expectation is thus consistent with the conjecture that the scaling limit of critical $O(2)$ loop-decorated maps is described by the $\CLE_4$ together with an independent critical Liouville quantum gravity (see e.g.\@ Conjecture 2.1 for a precise conjecture).

— The conformally invariant metric on CLE$_4$ III: uniqueness  (2609.04140 - Kammerer et al., 3 Sep 2026) in Section 1, Introduction, subsection “Overview”

This makes our expectation agree with the conjecture that the scaling limit of critical $O(2)$ loop-decorated maps is described by the $\CLE_4$ together with an independent critical Liouville quantum gravity (see, e.g.,Conjecture 2.1 for a precise conjecture).

— The conformally invariant metric on CLE$_4$ I: subsequential limits of the non-simple CLE graph metric  (2609.04138 - Kammerer et al., 3 Sep 2026) in Section 1, Subsection 1.1 (Random planar map motivation)