Critical Properties of Two-dimensional Simplicial Quantum Gravity

Simplicial higher-derivative quantum gravity is investigated in two dimensions for a manifold of toroidal topology. The manifold is dynamically triangulated using Regge’s formulation of gravity, with continuously varying edge lengths and fiied coordination number. Critical exponents are estimated by computer simulation on lattices with up to 786432 edges, and compared to the continuum conformal field theory results for central charge zero (pure gravity), one haif (Ising model coupled to gravity), one and two (massless scalar field coupled to gravity). The dependence of critical properties on the coefficient of the curvature squared term and the gravitational functional measure is investigated, suggesting universal critical behavior at least within a certain class of measures. In the case of pure gravity, we have computed the string susceptibility exponent for both the torus and the sphere, and our estimates agree with the exact result of KPZ. The fluctuations in the area density are consistent with the behavior expected for a massless scalar field, the Liouviiie mode. In the case of gravity coupled to a massless scalar field, we have computed what corresponds to the fractal dimension of the surface, and found it to be infinite. The critical exponents associated with the Ising model coupled to gravity on a torus are found to be the same as for the Ising model in flat space.