Exact limiting variance constant for the critical discrepancy

Determine whether the normalized discrepancy variance for the infinite FK(4)-decorated planar map satisfies $\log^2(n)\operatorname{Var}(D_n)/n\to 4\pi^2$ as $n\to\infty$.

Background

The paper proves only two-sided bounds showing that Var(D_n) is of order n/log2(n), with normalized liminf at least 4pi2 and limsup at most 8pi2. The authors explicitly state their belief that the sharper constant 4pi2 is the true limit, but do not establish it because it is unnecessary for the principal functional scaling-limit theorem.

References

We believe that $\frac{\log2(n)}{n}Var(D_n)\to 4\pi2$ (that is, the covariance of the pair on the left-hand side of eq: main converges to the covariance of the limit), but we have not yet pursued this direction since it is not needed for our main result.

eq: main:

(S⌊nt⌋n, log⁡n2πn D⌊nt⌋)t∈R⟶d(Bt1, Bt2)t∈R\left( \frac{\mathcal{S}_{\lfloor nt\rfloor}}{\sqrt{n}},\ \frac{\log n}{2\pi\sqrt{n}}\,\mathcal{D}_{\lfloor nt\rfloor} \right)_{t\in R} \overset{\textnormal{d}}{\longrightarrow} \left(B^1_t,\, B^2_{t}\right)_{t\in R}

— Scaling limits of critical FK-decorated random planar maps with $q=4$  (2511.21480 - Silva et al., 26 Nov 2025) in Section 1, immediately after Theorem 1.2 (Variance estimate)