5,191 research outputs found
Two-Dimensional Quantum Geometry
In these lectures we review our present understanding of the fractal
structure of two-dimensional Euclidean quantum gravity coupled to matter.Comment: Lectures presented at "The 53rd Cracow School of Theoretical Physics:
Conformal Symmetry and Perspectives in Quantum and Mathematical Gravity",
June 28 - July 7, 2013, Zakopane, Polan
In Search of Fundamental Discreteness in 2+1 Dimensional Quantum Gravity
Inspired by previous work in 2+1 dimensional quantum gravity, which found
evidence for a discretization of time in the quantum theory, we reexamine the
issue for the case of pure Lorentzian gravity with vanishing cosmological
constant and spatially compact universes of genus larger than 1. Taking as our
starting point the Chern-Simons formulation with Poincare gauge group, we
identify a set of length variables corresponding to space- and timelike
distances along geodesics in three-dimensional Minkowski space. These are Dirac
observables, that is, functions on the reduced phase space, whose quantization
is essentially unique. For both space- and timelike distance operators, the
spectrum is continuous and not bounded away from zero.Comment: 29 pages, 18 figure
The effective kinetic term in CDT
We report on recently performed simulations of Causal Dynamical
Triangulations (CDT) in 2+1 dimensions aimed at studying its effective dynamics
in the continuum limit. Two pieces of evidence from completely different
measurements are presented suggesting that three-dimensional CDT is effectively
described by an action with kinetic term given by a modified Wheeler-De Witt
metric. These observations could strengthen an earlier observed connection
between CDT and Horava-Lifshitz gravity. One piece of evidence comes from
measurements of the modular parameter in CDT simulations with spatial topology
of a torus, the other from measurements of local metric fluctuations close to a
fixed spatial boundary.Comment: 4 pages, 4 figures, based on a talk given at Loops '11, Madrid, to
appear in Journal of Physics: Conference Series (JPCS
Generalized multicritical one-matrix models
We show that there exists a simple generalization of Kazakov's multicritical
one-matrix model, which interpolates between the various multicritical points
of the model. The associated multicritical potential takes the form of a power
series with a heavy tail, leading to a cut of the potential and its derivative
at the real axis, and reduces to a polynomial at Kazakov's multicritical
points. From the combinatorial point of view the generalized model allows
polygons of arbitrary large degrees (or vertices of arbitrary large degree,
when considering the dual graphs), and it is the weight assigned to these large
order polygons which brings about the interpolation between the multicritical
points in the one-matrix model.Comment: 25 page
Scale-dependent Hausdorff dimensions in 2d gravity
By appropriate scaling of coupling constants a one-parameter family of
ensembles of two-dimensional geometries is obtained, which interpolates between
the ensembles of (generalized) causal dynamical triangulations and ordinary
dynamical triangulations. We study the fractal properties of the associated
continuum geometries and identify both global and local Hausdorff dimensions.Comment: 12 pages, 3 figure
Multilayered folding with voids
In the deformation of layered materials such as geological strata, or stacks
of paper, mechanical properties compete with the geometry of layering. Smooth,
rounded corners lead to voids between the layers, while close packing of the
layers results in geometrically-induced curvature singularities. When voids are
penalized by external pressure, the system is forced to trade off these
competing effects, leading to sometimes striking periodic patterns.
In this paper we construct a simple model of geometrically nonlinear
multi-layered structures under axial loading and pressure confinement, with
non-interpenetration conditions separating the layers. Energy minimizers are
characterized as solutions of a set of fourth-order nonlinear differential
equations with contact-force Lagrange multipliers, or equivalently of a
fourth-order free-boundary problem. We numerically investigate the solutions of
this free boundary problem, and compare them with the periodic solutions
observed experimentally
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