Geometry of a two-dimensional quantum gravity: numerical study

Abstract A two-dimensional quantum gravity is simulated by means of the dynamical triangulation model. The size of the lattice was up to hundred thousand triangles. Massively parallel simulations and recursive sampling were implemented independently and produced similar results. Wherever the analytical predictions existed, our results confirmed them. The cascade process of baby universes formulation a la Coleman-Hawking scenario in a two-dimensional case has been observed. We observed that there is a simple universal inclusive probability for a baby universe to appear. This anomalous branching of surfaces led to a rapid growth of the integral curvature inside a circle. The volume of a disk in the internal metric has been proven to grow faster than any power of radius. The scaling prediction for the mean square extent given by the Liouville theory has been confirmed. However, the naive expectation for the average Liouville lagrangian 〈∫(▽ φ ) 2 〉 is about 1 order of magnitude different from the results. This apparently points out to some flaws in the current definition of a Liouville model.

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