Determine the Hausdorff dimension of noncritical LQG metric spaces

Determine the value of the Hausdorff dimension d_gamma of the gamma-Liouville quantum gravity metric space for gamma not equal to sqrt(8/3), thereby addressing a major open problem in Liouville quantum gravity theory.

Background

The paper situates directed distances in bipolar-oriented triangulations within the broader universality theory of Liouville quantum gravity. For the special parameter gamma = sqrt(8/3), the Hausdorff dimension of the LQG metric space is known, but the corresponding dimension is not known for other values of gamma. The authors identify its computation as a central unresolved problem in LQG theory.

References

However, the value of $d_\gamma$ is unknown for $\gamma\not=\sqrt{8/3}$ (we have $d_{\sqrt{8/3}}=4$), and computing it is one of the most important open problems in LQG theory.

— Directed distances in bipolar-oriented triangulations: exact exponents and scaling limits  (2510.26123 - Borga et al., 30 Oct 2025) in Section 1, Overview