In critical as well as in non-critical string theory the partition function reduces to an integral over moduli space after integration over matter fields. For non-critical string theory this moduli integrand is known for genus one surfaces. The formalism of dynamical triangulations provides us with a regularization of non-critical string theory. We show how to assign in a simple and geometrical way a moduli parameter to each triangulation. After integrating over possible matter fields we can thus construct the moduli integrand. We show numerically for $c=0$ and $c=-2$ non-critical strings that the moduli integrand converges to the known continuum expression when the number of triangles goes to infinity.Comment: 32 pages, many (nice) figure

The behaviour of baby universes has been an important ingredient in understanding and quantifying non-critical string theory or, equivalently, models of twodimensional Euclidean quantum gravity coupled to matter. Within a regularized description based on dynamical triangulations, we amend an earlier conjecture by Jain and Mathur on the scaling behaviour of genus-g surfaces containing particular baby universe 'necks', and perform a nontrivial numerical check on our improved conjecture.

Two-dimensional quantum gravity, defined either via scaling limits of random discrete surfaces or via Liouville quantum gravity, is known to possess a geometry that is genuinely fractal with a Hausdorff dimension equal to 4. Coupling gravity to a statistical system at criticality changes the fractal properties of the geometry in a way that depends on the central charge of the critical system. Establishing the dependence of the Hausdorff dimension on this central charge c has been an important open problem in physics and mathematics in the past decades. All simulation data produced thus far has supported a formula put forward by Watabiki in the nineties. However, recent rigorous bounds on the Hausdorff dimension in Liouville quantum gravity show that Watabiki's formula cannot be correct when c approacheś 8. Based on simulations of discrete surfaces encoded by random planar maps and a numerical implementation of Liouville quantum gravity, we obtain new finite-size scaling estimates of the Hausdorff dimension that are in clear contradiction with Watabiki's formula for all simulated values of c P p´8, 0q. Instead, the most reliable data in the range c P r´12.5, 0q is in very good agreement with an alternative formula that was recently suggested by Ding and Gwynne. The estimates for c P p´8,´12.5q display a negative deviation from the latter formula, but the scaling is seen to be less accurate in this regime.