Scaling limits of critical FK-decorated random planar maps with $q=4$

We establish the first scaling limit for FK($q$)-weighted planar maps in the critical case $q=4$, resolving a problem that has remained open since Sheffield's seminal work arXiv:1108.2241. In that work, Sheffield proved a scaling limit for $q<4$ via the celebrated hamburger-cheeseburger bijection, which initiated the peanosphere (mating-of-trees) approach to Liouville quantum gravity. We prove that, at criticality, the associated burger count $\mathcal{S}$ and discrepancy $\mathcal{D}$ satisfy $$\left(\frac{\mathcal{S}_{\lfloor nt \rfloor}}{\sqrt{n}}, \frac{\log(n)}{{2\pi }\sqrt{n}} \mathcal{D}_{\lfloor nt \rfloor}\right)_{t\in\mathbb{R}} \stackrel{\text{d}}{\longrightarrow} (B^1_t, B^2_{t})_{t\in\mathbb{R}},$$ where $B^1$ and $B^2$ are independent two-sided Brownian motions. To the best of our knowledge, no conjecture for the correct discrepancy scaling factor had previously been formulated. Matching the limiting process with the critical mating of trees arXiv:2109.00275, we establish the first rigorous planar map convergence towards CLE$_4$ and critical ($\gamma=2$) Liouville quantum gravity, in the peanosphere sense. Our proof is based on a novel approach that reveals the exactly solvable nature of the model through a correspondence with the (bicoloured) fully packed loop-$O(2)$ model on triangulations, and yields critical geometric exponents matching the predictions of conformal field theory.