We study 2D quantum gravity on spherical topologies employing the Regge calculus approach with the dl/l measure. Instead of the normally used fixed nonregular triangulations we study random triangulations which are generated by the standard Voronoi–Delaunay procedure. For each system size we average the results over four different realizations of the random lattices. We compare both types of triangulations quantitatively and investigate how the difference in the expectation value of the squared curvature, R2, for fixed and random triangulations depends on the lattice size and the surface area A. We try to measure the string susceptibility exponents through finite-size scaling analyses of the expectation value of an added R2 interaction term, using two conceptually quite different procedures. The approach, where an ultraviolet cutoff is held fixed in the scaling limit, is found to be plagued with inconsistencies, as has already previously been pointed out by us. In a conceptually different approach, where the area A is held fixed, these problems are not present. We find the string susceptibility exponent γ′ str in rough agreement with theoretical predictions for the sphere, whereas the estimate for γ str appears to be too negative. However, our results are hampered by the presence of severe finite-size corrections to scaling, which lead to systematic uncertainties well above our statistical errors. We feel that the present methods of estimating the string susceptibilities by finite-size scaling studies are not accurate enough to serve as testing grounds to decide the success or failure of quantum Regge calculus.
Shape dependence of the finite-size scaling limit in a strongly anisotropic $\mathsf{O(\infty)}$ model