THE HAUSDORFF DIMENSION OF SURFACES IN 2-DIMENSIONAL QUANTUM-GRAVITY COUPLED TO ISING MINIMAL MATTER

Within the framework of string field theory the intrinsic Hausdorff di mension d(H) of the ensemble of surfaces in two-dimensional quantum gr avity has recently been claimed to be 2m for the class of unitary mini mal models (p = m+1,q = m). This contradicts recent results from numer ical simulations, which consistently find d(H) approximate to 4 in the cases m = 2, 3, 5 and infinity. The string field calculations rely on identifying the scaling behavior of geodesic distance and area with r espect 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 study the relationship between the mean area and the boundar y length for pure gravity and the Ising model coupled to gravity. We d iscuss how this relationship is modified by relevant perturbations in the Ising model. We discuss how this leads to a modified value for the Hausdorff dimension. (C) 1997 Published by Elsevier Science B.V.