Extend directed LQG metrics beyond theta = 2pi

Determine whether a directed variant of the Liouville quantum gravity metric can be defined for parameter values theta greater than 2pi.

Background

The directed LQG framework in the paper is parametrized by gamma and theta, with theta = 2pi corresponding to the undirected metric. For gamma greater than sqrt(8/3), the distinguished parameter theta_*(gamma) exceeds 2pi, raising the question of whether the directed-metric construction extends to that regime.

References

When $\gamma > \sqrt{8/3}$, we have $\theta_*(\gamma) > 2\pi$. We do not know whether it is possible to make sense of a directed variant of the LQG metric with $\theta > 2\pi$.

— Directed distances in bipolar-oriented triangulations: exact exponents and scaling limits  (2510.26123 - Borga et al., 30 Oct 2025) in Section 2.3, Directed LQG metrics; Figure 3 caption