42 research outputs found
Scaling and the Fractal Geometry of Two-Dimensional Quantum Gravity
We examine the scaling of geodesic correlation functions in two-dimensional
gravity and in spin systems coupled to gravity. The numerical data support the
scaling hypothesis and indicate that the quantum geometry develops a
non-perturbative length scale. The existence of this length scale allows us to
extract a Hausdorff dimension. In the case of pure gravity we find d_H approx.
3.8, in support of recent theoretical calculations that d_H = 4. We also
discuss the back-reaction of matter on the geometry.Comment: 16 pages, LaTeX format, 8 eps figure
Ising Model Coupled to Three-Dimensional Quantum Gravity
We have performed Monte Carlo simulations of the Ising model coupled to
three-dimensional quantum gravity based on a summation over dynamical
triangulations. These were done both in the microcanonical ensemble, with the
number of points in the triangulation and the number of Ising spins fixed, and
in the grand canoncal ensemble. We have investigated the two possible cases of
the spins living on the vertices of the triangulation (``diect'' case) and the
spins living in the middle of the tetrahedra (``dual'' case). We observed phase
transitions which are probably second order, and found that the dual
implementation more effectively couples the spins to the quantum gravity.Comment: 11 page
Three Dimensional Quantum Gravity Coupled to Ising Matter
We establish the phase diagram of three--dimensional quantum gravity coupled
to Ising matter. We find that in the negative curvature phase of the quantum
gravity there is no disordered phase for ferromagnetic Ising matter because the
coordination number of the sites diverges. In the positive curvature phase of
the quantum gravity there is evidence for two spin phases with a first order
transition between them.Comment: 12 page
Three-Dimensional Quantum Gravity Coupled to Gauge Fields
We show how to simulate U(1) gauge fields coupled to three-dimensional
quantum gravity and then examine the phase diagram of this system. Quenched
mean field theory suggests that a transition separates confined and deconfined
phases (for the gauge matter) in both the negative curvature phase and the
positive curvature phase of the quantum gravity, but numerical simulations find
no evidence for such transitions.Comment: 16 page
Numerical simulation of stochastic vortex tangles
We present the results of simulation of the chaotic dynamics of quantized
vortices in the bulk of superfluid He II.
Evolution of vortex lines is calculated on the base of the Biot-Savart law.
The dissipative effects appeared from the interaction with the normal
component, or/and from relaxation of the order parameter are taken into
account. Chaotic dynamics appears in the system via a random forcing, e.i. we
use the Langevin approach to the problem. In the present paper we require the
correlator of the random force to satisfy the fluctuation-disspation relation,
which implies that thermodynamic equilibrium should be reached. In the paper we
describe the numerical methods for integration of stochastic differential
equation (including a new algorithm for reconnection processes), and we present
the results of calculation of some characteristics of a vortex tangle such as
the total length, distribution of loops in the space of their length, and the
energy spectrum.Comment: 8 pages, 5 figure
On the Absence of an Exponential Bound in Four Dimensional Simplicial Gravity
We have studied a model which has been proposed as a regularisation for four
dimensional quantum gravity. The partition function is constructed by
performing a weighted sum over all triangulations of the four sphere. Using
numerical simulation we find that the number of such triangulations containing
simplices grows faster than exponentially with . This property ensures
that the model has no thermodynamic limit.Comment: 8 pages, 2 figure
Phase Structure of Dynamical Triangulation Models in Three Dimensions
The dynamical triangulation model of three-dimensional quantum gravity is
shown to have a line of transitions in an expanded phase diagram which includes
a coupling mu to the order of the vertices. Monte Carlo renormalization group
and finite size scaling techniques are used to locate and characterize this
line. Our results indicate that for mu < mu1 ~ -1.0 the model is always in a
crumpled phase independent of the value of the curvature coupling. For mu < 0
the results are in agreement with an approximate mean field treatment. We find
evidence that this line corresponds to first order transitions extending to
positive mu. However, the behavior appears to change for mu > mu2 ~ 2-4. The
simplest scenario that is consistent with the data is the existence of a
critical end point
An Effective Model for Crumpling in Two Dimensions?
We investigate the crumpling transition for a dynamically triangulated random
surface embedded in two dimensions using an effective model in which the
disordering effect of the variables on the correlations of the normals is
replaced by a long-range ``antiferromagnetic'' term. We compare the results
from a Monte Carlo simulation with those obtained for the standard action which
retains the 's and discuss the nature of the phase transition.Comment: 5 page
Gauge Invariance in Simplicial Gravity
The issue of local gauge invariance in the simplicial lattice formulation of
gravity is examined. We exhibit explicitly, both in the weak field expansion
about flat space, and subsequently for arbitrarily triangulated background
manifolds, the exact local gauge invariance of the gravitational action, which
includes in general both cosmological constant and curvature squared terms. We
show that the local invariance of the discrete action and the ensuing zero
modes correspond precisely to the diffeomorphism invariance in the continuum,
by carefully relating the fundamental variables in the discrete theory (the
edge lengths) to the induced metric components in the continuum. We discuss
mostly the two dimensional case, but argue that our results have general
validity. The previous analysis is then extended to the coupling with a scalar
field, and the invariance properties of the scalar field action under lattice
diffeomorphisms are exhibited. The construction of the lattice conformal gauge
is then described, as well as the separation of lattice metric perturbations
into orthogonal conformal and diffeomorphism part. The local gauge invariance
properties of the lattice action show that no Fadeev-Popov determinant is
required in the gravitational measure, unless lattice perturbation theory is
performed with a gauge-fixed action, such as the one arising in the lattice
analog of the conformal or harmonic gauges.Comment: LaTeX, 68 pages, 24 figure
Ising-link Quantum Gravity
We define a simplified version of Regge quantum gravity where the link
lengths can take on only two possible values, both always compatible with the
triangle inequalities. This is therefore equivalent to a model of Ising spins
living on the links of a regular lattice with somewhat complicated, yet local
interactions. The measure corresponds to the natural sum over all 2^links
configurations, and numerical simulations can be efficiently implemented by
means of look-up tables. In three dimensions we find a peak in the ``curvature
susceptibility'' which grows with increasing system size. However, the value of
the corresponding critical exponent as well as the behavior of the curvature at
the transition differ from that found by Hamber and Williams for the Regge
theory with continuously varying link lengths.Comment: 11 page