Scaling limit of self-dual FK-weighted planar maps to Liouville quantum gravity

Establish that suitably renormalised and conformally embedded self-dual FK(q)-weighted planar maps converge to a gamma-Liouville quantum gravity surface, with q = 2 + 2 cos(pi gamma^2/2) and gamma in (sqrt(2),2).

Background

The paper describes a principal conjecture connecting self-dual FK(q)-weighted planar maps to Liouville quantum gravity. The conjecture concerns convergence after both geometric renormalisation and conformal embedding, and specifies the relationship between the FK parameter q and the LQG parameter gamma.

Although the paper proves exact enumerative and tail-asymptotic results and discusses evidence from inventory accumulation and peanosphere convergence, it does not establish convergence in the full geometric sense stated here. The question is therefore explicitly unresolved in the paper.

References

Furthermore, one of the principal conjectures in the field -- closely related to the DDK ansatz mentioned above -- states that, when suitably renormalised and conformally embedded into the Riemann sphere, self-dual FK$(q)$-weighted planar maps converge to a $\gamma$-LQG surface, where

q = 2 + 2\cos\left(\frac{\pi \gamma2}{2}\right) \quad \text{and} \quad \gamma\in(\sqrt{2},2).

— Critical behaviour of the fully packed loop-$O(n)$ model on planar triangulations  (2512.05867 - Berestycki et al., 5 Dec 2025) in Section 1, Introduction

Confirming this expectation by rigorous mathematical results is a major open problem, and gaining a better combinatorial understanding of hypermaps will be helpful in such a task.

— Enumeration of plane hypermaps with a mixed boundary I  (2608.19947 - Bouttier et al., 20 Aug 2026) in Section 1, subsection “Context and motivations”