Joint convergence of future-exploration burger counts

Establish joint convergence in distribution for the hamburger and cheeseburger counts along the exploration into the future, and thereby upgrade the tightness of their discrepancy difference to proper convergence in distribution.

Background

For the future exploration, the paper proves marginal stable-limit statements for the hamburger and cheeseburger counts and uses them to obtain tightness of the rescaled discrepancy difference. Because the two counts are not independent, marginal convergence does not by itself identify their joint limit. The authors explicitly indicate that a joint Laplace-transform analysis could resolve this remaining issue, but they do not carry it out.

References

We believe that the proof of \cref{prop:Laplace_H_PF} could be adapted to deal with the joint Laplace transform of $\mathcal{H}*(P_{F})$ and $\mathcal{C}*(P_{F})$, which would then upgrade eq: scaling to stable HPF to a joint convergence statement and eq: scaling to stable DPF to proper convergence in distribution.

eq: scaling to stable HPF:

n⟶dζ^,_n\stackrel{\textnormal{d}}{\longrightarrow} \hat\zeta,

eq: scaling to stable DPF:

(1nlog⁡n∑k=1nD∗(PFk),  n≥1)=(n−nnlog⁡n,  n≥1)\left(\frac{1}{n\log n}\sum^n_{k=1} \mathcal{D}^*(P_{F}^k), \; n\geq 1 \right) = \left(\frac{_n-_n}{n\log n}, \; n\geq 1 \right)

— Scaling limits of critical FK-decorated random planar maps with $q=4$  (2511.21480 - Silva et al., 26 Nov 2025) in Remark following Theorem 6.1 (Scaling limits in the exploration into the future)