1,051 research outputs found
Face pairing graphs and 3-manifold enumeration
The face pairing graph of a 3-manifold triangulation is a 4-valent graph
denoting which tetrahedron faces are identified with which others. We present a
series of properties that must be satisfied by the face pairing graph of a
closed minimal P^2-irreducible triangulation. In addition we present
constraints upon the combinatorial structure of such a triangulation that can
be deduced from its face pairing graph. These results are then applied to the
enumeration of closed minimal P^2-irreducible 3-manifold triangulations,
leading to a significant improvement in the performance of the enumeration
algorithm. Results are offered for both orientable and non-orientable
triangulations.Comment: 30 pages, 57 figures; v2: clarified some passages and generalised the
final theorem to the non-orientable case; v3: fixed a flaw in the proof of
the conical face lemm
Spacetime as a Feynman diagram: the connection formulation
Spin foam models are the path integral counterparts to loop quantized
canonical theories. In the last few years several spin foam models of gravity
have been proposed, most of which live on finite simplicial lattice spacetime.
The lattice truncates the presumably infinite set of gravitational degrees of
freedom down to a finite set. Models that can accomodate an infinite set of
degrees of freedom and that are independent of any background simplicial
structure, or indeed any a priori spacetime topology, can be obtained from the
lattice models by summing them over all lattice spacetimes. Here we show that
this sum can be realized as the sum over Feynman diagrams of a quantum field
theory living on a suitable group manifold, with each Feynman diagram defining
a particular lattice spacetime. We give an explicit formula for the action of
the field theory corresponding to any given spin foam model in a wide class
which includes several gravity models. Such a field theory was recently found
for a particular gravity model [De Pietri et al, hep-th/9907154]. Our work
generalizes this result as well as Boulatov's and Ooguri's models of three and
four dimensional topological field theories, and ultimately the old matrix
models of two dimensional systems with dynamical topology. A first version of
our result has appeared in a companion paper [gr-qc\0002083]: here we present a
new and more detailed derivation based on the connection formulation of the
spin foam models.Comment: 32 pages, 2 figure
The geometry of dynamical triangulations
We discuss the geometry of dynamical triangulations associated with
3-dimensional and 4-dimensional simplicial quantum gravity. We provide
analytical expressions for the canonical partition function in both cases, and
study its large volume behavior. In the space of the coupling constants of the
theory, we characterize the infinite volume line and the associated critical
points. The results of this analysis are found to be in excellent agreement
with the MonteCarlo simulations of simplicial quantum gravity. In particular,
we provide an analytical proof that simply-connected dynamically triangulated
4-manifolds undergo a higher order phase transition at a value of the inverse
gravitational coupling given by 1.387, and that the nature of this transition
can be concealed by a bystable behavior. A similar analysis in the
3-dimensional case characterizes a value of the critical coupling (3.845) at
which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil
Ricci Curvature of the Internet Topology
Analysis of Internet topologies has shown that the Internet topology has
negative curvature, measured by Gromov's "thin triangle condition", which is
tightly related to core congestion and route reliability. In this work we
analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin,
etc. Ricci curvature measures whether local distances diverge or converge. It
is a more local measure which allows us to understand the distribution of
curvatures in the network. We show by various Internet data sets that the
distribution of Ricci cuvature is spread out, suggesting the network topology
to be non-homogenous. We also show that the Ricci curvature has interesting
connections to both local measures such as node degree and clustering
coefficient, global measures such as betweenness centrality and network
connectivity, as well as auxilary attributes such as geographical distances.
These observations add to the richness of geometric structures in complex
network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201
The persistent cosmic web and its filamentary structure I: Theory and implementation
We present DisPerSE, a novel approach to the coherent multi-scale
identification of all types of astrophysical structures, and in particular the
filaments, in the large scale distribution of matter in the Universe. This
method and corresponding piece of software allows a genuinely scale free and
parameter free identification of the voids, walls, filaments, clusters and
their configuration within the cosmic web, directly from the discrete
distribution of particles in N-body simulations or galaxies in sparse
observational catalogues. To achieve that goal, the method works directly over
the Delaunay tessellation of the discrete sample and uses the DTFE density
computed at each tracer particle; no further sampling, smoothing or processing
of the density field is required.
The idea is based on recent advances in distinct sub-domains of computational
topology, which allows a rigorous application of topological principles to
astrophysical data sets, taking into account uncertainties and Poisson noise.
Practically, the user can define a given persistence level in terms of
robustness with respect to noise (defined as a "number of sigmas") and the
algorithm returns the structures with the corresponding significance as sets of
critical points, lines, surfaces and volumes corresponding to the clusters,
filaments, walls and voids; filaments, connected at cluster nodes, crawling
along the edges of walls bounding the voids. The method is also interesting as
it allows for a robust quantification of the topological properties of a
discrete distribution in terms of Betti numbers or Euler characteristics,
without having to resort to smoothing or having to define a particular scale.
In this paper, we introduce the necessary mathematical background and
describe the method and implementation, while we address the application to 3D
simulated and observed data sets to the companion paper.Comment: A higher resolution version is available at
http://www.iap.fr/users/sousbie together with complementary material.
Submitted to MNRA
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