Directed critical Liouville quantum gravity metric space

Construct the directed version of the critical Liouville quantum gravity metric space associated with the Brownian separable permuton and prove that its Hausdorff dimension equals α(p)^{-1}.

Background

The paper connects longest directed paths in series–parallel maps with longest increasing subsequences and suggests a continuum interpretation through critical Liouville quantum gravity and SLE4/CLE4 structures.

A directed critical Liouville quantum gravity metric space has not yet been constructed in the required form. The authors further conjecture that its Hausdorff dimension should be the reciprocal of the exponent α(p).

References

In this correspondence, it is conjectured that Brownian separable permutons can be constructed using $\text{SLE}_4/\text{CLE}_4$ and critical Liouville quantum gravity ($\gamma=2$). In connection to the aforementioned discrete picture, we expect the exponent $\alpha(p){-1}$ to coincide with the Hausdorff dimension of a directed version of the critical Liouville quantum gravity metric space; a space whose construction still represents a major challenge but which, at least in the critical regime, appears significantly more approachable in light of the results presented in this paper.

— The longest increasing subsequence of Brownian separable permutons  (2506.19123 - Adhikari et al., 23 Jun 2025) in Section 1.3, paragraph “A directed version of the Liouville quantum gravity metric”