Geodesic distances in Liouville quantum gravity

Understand

In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice.

• This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d quantum gravity on the torus coupled to conformal matter fields, showing agreement with a conjectured formula by Y.
• Watabiki.
• Finally, the numerical tools are put to test by quantitatively comparing the distribution of lengths of shortest cycles to the corresponding distribution in large random triangulations.