3,936 research outputs found
Uniform Infinite Planar Triangulations
The existence of the weak limit as n --> infinity of the uniform measure on
rooted triangulations of the sphere with n vertices is proved. Some properties
of the limit are studied. In particular, the limit is a probability measure on
random triangulations of the plane.Comment: 36 pages, 4 figures; Journal revised versio
Unimodular lattice triangulations as small-world and scale-free random graphs
Real-world networks, e.g. the social relations or world-wide-web graphs,
exhibit both small-world and scale-free behaviour. We interpret lattice
triangulations as planar graphs by identifying triangulation vertices with
graph nodes and one-dimensional simplices with edges. Since these
triangulations are ergodic with respect to a certain Pachner flip, applying
different Monte-Carlo simulations enables us to calculate average properties of
random triangulations, as well as canonical ensemble averages using an energy
functional that is approximately the variance of the degree distribution. All
considered triangulations have clustering coefficients comparable with real
world graphs, for the canonical ensemble there are inverse temperatures with
small shortest path length independent of system size. Tuning the inverse
temperature to a quasi-critical value leads to an indication of scale-free
behaviour for degrees . Using triangulations as a random graph model
can improve the understanding of real-world networks, especially if the actual
distance of the embedded nodes becomes important.Comment: 17 pages, 6 figures, will appear in New J. Phy
Irreducible triangulations of surfaces with boundary
A triangulation of a surface is irreducible if no edge can be contracted to
produce a triangulation of the same surface. In this paper, we investigate
irreducible triangulations of surfaces with boundary. We prove that the number
of vertices of an irreducible triangulation of a (possibly non-orientable)
surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was
known only for surfaces without boundary (b=0). While our technique yields a
worse constant in the O(.) notation, the present proof is elementary, and
simpler than the previous ones in the case of surfaces without boundary
The polytope of non-crossing graphs on a planar point set
For any finite set \A of points in , we define a
-dimensional simple polyhedron whose face poset is isomorphic to the
poset of ``non-crossing marked graphs'' with vertex set \A, where a marked
graph is defined as a geometric graph together with a subset of its vertices.
The poset of non-crossing graphs on \A appears as the complement of the star
of a face in that polyhedron.
The polyhedron has a unique maximal bounded face, of dimension
where is the number of points of \A in the interior of \conv(\A). The
vertices of this polytope are all the pseudo-triangulations of \A, and the
edges are flips of two types: the traditional diagonal flips (in
pseudo-triangulations) and the removal or insertion of a single edge.
As a by-product of our construction we prove that all pseudo-triangulations
are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has
been reshape
Quantum Gravity and Matter: Counting Graphs on Causal Dynamical Triangulations
An outstanding challenge for models of non-perturbative quantum gravity is
the consistent formulation and quantitative evaluation of physical phenomena in
a regime where geometry and matter are strongly coupled. After developing
appropriate technical tools, one is interested in measuring and classifying how
the quantum fluctuations of geometry alter the behaviour of matter, compared
with that on a fixed background geometry.
In the simplified context of two dimensions, we show how a method invented to
analyze the critical behaviour of spin systems on flat lattices can be adapted
to the fluctuating ensemble of curved spacetimes underlying the Causal
Dynamical Triangulations (CDT) approach to quantum gravity. We develop a
systematic counting of embedded graphs to evaluate the thermodynamic functions
of the gravity-matter models in a high- and low-temperature expansion. For the
case of the Ising model, we compute the series expansions for the magnetic
susceptibility on CDT lattices and their duals up to orders 6 and 12, and
analyze them by ratio method, Dlog Pad\'e and differential approximants. Apart
from providing evidence for a simplification of the model's analytic structure
due to the dynamical nature of the geometry, the technique introduced can shed
further light on criteria \`a la Harris and Luck for the influence of random
geometry on the critical properties of matter systems.Comment: 40 pages, 15 figures, 13 table
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