Minimal Dynamical Triangulations of Random Surfaces

We introduce and investigate numerically a minimal class of dynamical triangulations of two-dimensional gravity on the sphere in which only vertices of order five, six or seven are permitted. We show firstly that this restriction of the local coordination number, or equivalently intrinsic scalar curvature, leaves intact the fractal structure characteristic of generic dynamically triangulated random surfaces. Furthermore the Ising model coupled to minimal twodimensional gravity still possesses a continuous phase transition. The critical exponents of this transition correspond to the usual KPZ exponents associated with coupling a central charge c = 1 2 model to two-dimensional gravity. That two-dimensional (2D) quantum gravity can be regularized and studied using dynamical triangulations (DT) is well known [1] and has lead to extensive study of the properties of dynamically triangulated random surfaces (DTRS). Indeed, many of the properties of critical spin systems on such lattices have been shown to follow from continuum treatments of central charge c < 1 theories coupled to 2D-quantum gravity. Furthermore, this formulation has the advantage that it renders tractable the calculation of many features of these theories which are inaccessible to continuum methods. Foremost amongst these are questions related to the quantum geometry and fractal structure of the typical 2D manifolds appearing in the partition function [2]. It has often been speculated that some of these models should also find a realization in condensed matter physics as models of membranes and/or interfaces with fluid in-plane degrees of freedom. The problem in trying to map such problems onto dynamically triangulated models, however, has always been the occurrence of large vertex coordination numbers in an arbitrary random graph – real condensed matter