The Hausdorff Dimension of Surfaces in Two-Dimensional Quantum Gravity Coupled to Unitary Minimal Matter

Within the framework of string field theory the intrinsic Hausdorff dimension $d_H$ of the ensemble of surfaces in two-dimensional quantum gravity has recently been claimed to be $2m$ for the class of unitary minimal models $(p=m+1,q=m)$. This contradicts recent results from numerical simulations, which consistently find $d_H \approx 4$ in the cases $m=2$, 3, 5 and $\infty$. The string field calculations rely on identifying the scaling behavior of geodesic distance and area with respect to a common length scale $l$. This length scale is introduced by formulating the models on a disk with fixed boundary length $l$. In this paper we show that there exists a scaling limit in which the relation between the mean area and the boundary length of the disk is such that $d_H = 4$ for all values of $m$. Furthermore we argue that this scaling limit is the required one to reproduce the continuum behavior of matter coupled to two-dimensional gravity.