Fixed versus random triangulations in 2D simplicial Regge calculus

We study 2D quantum gravity on spherical topologies using the Regge calculus approach with the $dl\mathrm{/}l$ measure. Instead of a fixed non-regular triangulation which has been used before, we study for each system size four different random triangulations, which are obtained according to the standard Voronoi-Delaunay procedure. We compare both approaches quantitatively and show that the difference in the expectation value of $R^2$ between the fixed and the random triangulation depends on the lattice size and the surface area $A$. We also try again to measure the string susceptibility exponents through a finite-size scaling Ansatz in the expectation value of an added $R^2$ interaction term in an approach where $A$ is held fixed. The string susceptibility exponent ${\gamma }_{str}^{\mathrm{\prime }}$ is shown to agree with theoretical predictions for the sphere, whereas the estimate for ${\gamma }_{str}$ appears to be too negative.