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 dH 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 dH ≈ 4 in the cases m = 2, 3, 5 and ∞. 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 dH = 4 for all values ofm. Furthermore we argue that this scaling limit is the required one to reproduce the continuum behavior of matter coupled to two-dimensional gravity.