Minimal dynamical triangulations of random surfaces

Abstract

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 systems typically place a cut-off on the number of interactions allowed by the microscopic degrees of freedom. It is possible that such cut-offs are unimportant in the 1

Uniform Infinite Planar Triangulations

“…That form of the asymptotics is not dependent on the manifold, and is valid for any 2-dimensional manifold with or without boundaries. The same forms also appear when instead of triangulations other types of maps are considered, and was found to hold for a large variety of map types ( [11,10] and also the result of [13], related to our growth results). We therefore believe that many of the results here hold in a much more general context.…”

“…There exists a probability measure T2 (resp. T 3 ) supported on infinite planar triangulations of type 11 (resp. type Ill) such that Note.…”

“…This holds, for example, for planar triangulations with uniformly bounded degrees. However, for those distributions it is not clear how to prove that the limit exists (simulations support this [11] VEL parabolicity (for vertex extremal length) is a property of infinite graphs, closely related to circle packings for planar graphs. In graphs with bounded degrees it is equivalent to recurrence [21].…”

Uniform Infinite Planar Triangulations

“…This holds, for example, for planar triangulations with uniformly bounded degrees. However, for those distributions it is not clear how to prove that the limit exists (simulations support this [10] VEL parabolicity (for vertex extremal length) is a property of infinite graphs, closely related to circle packings for planar graphs. In graphs with bounded degrees it is equivalent to recurrence [21].…”

Three-dimensional simplicial gravity and degenerate triangulations

“…(2), although in two dimensions it is well known that while different ensembles yield the same continuum theory [7] the finite-size effects depend strongly the type of triangulations used. It has been shown that the less restricted the triangulations are -the larger the ensemble of triangulations isthe smaller the finite-size effects are [8,9]. This result can be understood intuitively as, for a given volume, with a larger triangulation-space it is easier to approximate a particular fractal structure.…”