118 research outputs found
Percolation on uniform infinite planar maps
We construct the uniform infinite planar map (UIPM), obtained as the n \to
\infty local limit of planar maps with n edges, chosen uniformly at random. We
then describe how the UIPM can be sampled using a "peeling" process, in a
similar way as for uniform triangulations. This process allows us to prove that
for bond and site percolation on the UIPM, the percolation thresholds are
p_c^bond=1/2 and p_c^site=2/3 respectively. This method also works for other
classes of random infinite planar maps, and we show in particular that for bond
percolation on the uniform infinite planar quadrangulation, the percolation
threshold is p_c^bond=1/3.Comment: 26 pages, 9 figure
Finding Hexahedrizations for Small Quadrangulations of the Sphere
This paper tackles the challenging problem of constrained hexahedral meshing.
An algorithm is introduced to build combinatorial hexahedral meshes whose
boundary facets exactly match a given quadrangulation of the topological
sphere. This algorithm is the first practical solution to the problem. It is
able to compute small hexahedral meshes of quadrangulations for which the
previously known best solutions could only be built by hand or contained
thousands of hexahedra. These challenging quadrangulations include the
boundaries of transition templates that are critical for the success of general
hexahedral meshing algorithms.
The algorithm proposed in this paper is dedicated to building combinatorial
hexahedral meshes of small quadrangulations and ignores the geometrical
problem. The key idea of the method is to exploit the equivalence between quad
flips in the boundary and the insertion of hexahedra glued to this boundary.
The tree of all sequences of flipping operations is explored, searching for a
path that transforms the input quadrangulation Q into a new quadrangulation for
which a hexahedral mesh is known. When a small hexahedral mesh exists, a
sequence transforming Q into the boundary of a cube is found; otherwise, a set
of pre-computed hexahedral meshes is used.
A novel approach to deal with the large number of problem symmetries is
proposed. Combined with an efficient backtracking search, it allows small
shellable hexahedral meshes to be found for all even quadrangulations with up
to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more
than 72 hexahedra. This algorithm is also used to find a construction to fill
arbitrary domains, thereby proving that any ball-shaped domain bounded by n
quadrangles can be meshed with no more than 78 n hexahedra. This very
significantly lowers the previous upper bound of 5396 n.Comment: Accepted for SIGGRAPH 201
Large Unicellular maps in high genus
We study the geometry of a random unicellular map which is uniformly
distributed on the set of all unicellular maps whose genus size is proportional
to the number of edges of the map. We prove that the distance between two
uniformly selected vertices of such a map is of order and the diameter
is also of order with high probability. We further prove that the map
is locally planar with high probability. The main ingredient of the proofs is
an exploration procedure which uses a bijection due to Chapuy, Feray and Fusy.Comment: 43 pages, 6 figures, revised file taking into account referee's
comment
Scaling limits for the uniform infinite quadrangulation
The uniform infinite planar quadrangulation is an infinite random graph
embedded in the plane, which is the local limit of uniformly distributed finite
quadrangulations with a fixed number of faces. We study asymptotic properties
of this random graph. In particular, we investigate scaling limits of the
profile of distances from the distinguished point called the root, and we get
asymptotics for the volume of large balls. As a key technical tool, we first
describe the scaling limit of the contour functions of the uniform infinite
well-labeled tree, in terms of a pair of eternal conditioned Brownian snakes.
Scaling limits for the uniform infinite quadrangulation can then be derived
thanks to an extended version of Schaeffer's bijection between well-labeled
trees and rooted quadrangulations.Comment: 36 page
A view from infinity of the uniform infinite planar quadrangulation
We introduce a new construction of the Uniform Infinite Planar
Quadrangulation (UIPQ). Our approach is based on an extension of the
Cori-Vauquelin-Schaeffer mapping in the context of infinite trees, in the
spirit of previous work. However, we release the positivity constraint on the
labels of trees which was imposed in these references, so that our construction
is technically much simpler. This approach allows us to prove the conjectures
of Krikun pertaining to the "geometry at infinity" of the UIPQ, and to derive
new results about the UIPQ, among which a fine study of infinite geodesics.Comment: 39 pages, 11 figure
The F model on dynamical quadrangulations
The dynamically triangulated random surface (DTRS) approach to Euclidean
quantum gravity in two dimensions is considered for the case of the elemental
building blocks being quadrangles instead of the usually used triangles. The
well-known algorithmic tools for treating dynamical triangulations in a Monte
Carlo simulation are adapted to the problem of these dynamical
quadrangulations. The thus defined ensemble of 4-valent graphs is appropriate
for coupling to it the 6- and 8-vertex models of statistical mechanics. Using a
series of extensive Monte Carlo simulations and accompanying finite-size
scaling analyses, we investigate the critical behaviour of the 6-vertex F model
coupled to the ensemble of dynamical quadrangulations and determine the matter
related as well as the graph related critical exponents of the model.Comment: LaTeX, 43 pages, 10 figures, 7 tables; substantially shortened and
revised version as published, for more details refer to V1, to be found at
http://arxiv.org/abs/hep-lat/0409028v
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