Baby Universes in 4d Dynamical Triangulation

We measure numerically the distribution of baby universes in the crumpled phase of the dynamical triangulation model of 4d quantum gravity. The relevance of the results to the issue of an exponential bound is discussed. The data are consistent with the existence of such a bound.Comment: 8 pages, 4 figure

Phase Structure of Four Dimensional Simplicial Quantum Gravity

We present the results of a high statistics Monte Carlo study of a model for four dimensional euclidean quantum gravity based on summing over triangulations. We show evidence for two phases; in one there is a logarithmic scaling on the mean linear extent with volume, whilst the other exhibits power law behaviour with exponent 1/2. We are able to extract a finite size scaling exponent governing the growth of the susceptibility peakComment: 11 pages (5 figures

A Conformal Hyperbolic Formulation of the Einstein Equations

We propose a re-formulation of the Einstein evolution equations that cleanly separates the conformal degrees of freedom and the non-conformal degrees of freedom with the latter satisfying a first order strongly hyperbolic system. The conformal degrees of freedom are taken to be determined by the choice of slicing and the initial data, and are regarded as given functions (along with the lapse and the shift) in the hyperbolic part of the evolution. We find that there is a two parameter family of hyperbolic systems for the non-conformal degrees of freedom for a given set of trace free variables. The two parameters are uniquely fixed if we require the system to be ``consistently trace-free'', i.e., the time derivatives of the trace free variables remains trace-free to the principal part, even in the presence of constraint violations due to numerical truncation error. We show that by forming linear combinations of the trace free variables a conformal hyperbolic system with only physical characteristic speeds can also be constructed.Comment: 4 page

The Renormalization Group and Dynamical Triangulations

A block spin renormalization group approach is introduced which can be applied to dynamical triangulations in any dimension.Comment: Talk presented at LATTICE96(gravity

Balls in Boxes and Quantum Gravity

Four dimensional simplicial gravity has been studied by means of Monte Carlo simulations for some time, the main outcome of the studies being that the model undergoes a discontinuous phase transition between an elongated and a crumpled phase when one changes the curvature (Newton) coupling. In the crumpled phase there are singular vertices growing extensively with the volume of the system whereas the elongated phase resembles a branched-polymer. We have postulated that this behaviour is a manifestation of the constrained-mean-field scenario as realised in the Branched Polymer or Balls-in-Boxes model. These models share all the features of 4D simplicial gravity except that they exhibit a continuous phase transition. We note here that this defect can be remedied by a suitable choice of ensemble.Comment: 3 pages, LaTeX, 2 figures, uses espcrc2.sty, talk given at LATTICE9

Singular Vertices and the Triangulation Space of the D-sphere

By a sequence of numerical experiments we demonstrate that generic triangulations of the $D-$sphere for $D>3$ contain one {\it singular} $(D-3)-$simplex. The mean number of elementary $D-$simplices sharing this simplex increases with the volume of the triangulation according to a simple power law. The lower dimension subsimplices associated with this $(D-3)-$simplex also show a singular behaviour. Possible consequences for the DT model of four-dimensional quantum gravity are discussed.Comment: 15 pages, 9 figure

Knot Invariants for Intersecting Loops

We generalize the braid algebra to the case of loops with intersections. We introduce the Reidemeister moves for 4 and 6-valent vertices to have a theory of rigid vertex equivalence. By considering representations of the extended braid algebra, we derive skein relations for link polynomials, which allow us to generalize any link Polynomial to the intersecting case. We perturbatively show that the HOMFLY Polynomials for intersecting links correspond to the vacuum expectation value of the Wilson line operator of the Chern Simon's Theory. We make contact with quantum gravity by showing that these polynomials are simply related with some solutions of the complete set of constraints with cosmological constantComment: 22 page

The Weak-Coupling Limit of 3D Simplicial Quantum Gravity

We investigate the weak-coupling limit, kappa going to infinity, of 3D simplicial gravity using Monte Carlo simulations and a Strong Coupling Expansion. With a suitable modification of the measure we observe a transition from a branched polymer to a crinkled phase. However, the intrinsic geometry of the latter appears similar to that of non-generic branched polymer, probable excluding the existence of a sensible continuum limit in this phase.Comment: 3 pages 4 figs. LATTICE99(Gravity

Condensation in the Backgammon model

We analyse the properties of a very simple ``balls-in-boxes'' model which can exhibit a phase transition between a fluid and a condensed phase, similar to behaviour encountered in models of random geometries in one, two and four dimensions. This model can be viewed as a generalisation of the backgammon model introduced by Ritort as an example of glassy behaviour without disorder.Comment: 14 pages, requires Latex2e + elsart.cls (supplied). 2 figures included as eps files. (some minor errors had been corrected and additional references added

Simplicial Gravity in Dimension Greater than Two

We consider two issues in the DT model of quantum gravity. First, it is shown that the triangulation space for D>3 is dominated by triangulations containing a single singular (D-3)-simplex composed of vertices with divergent dual volumes. Second we study the ergodicity of current simulation algorithms. Results from runs conducted close to the phase transition of the four-dimensional theory are shown. We see no strong indications of ergodicity br eaking in the simulation and our data support recent claims that the transition is most probably first order. Furthermore, we show that the critical properties of the system are determined by the dynamics of remnant singular vertices.Comment: Talk presented at LATTICE96(gravity