54,816 research outputs found
Random planar maps and graphs with minimum degree two and three
We find precise asymptotic estimates for the number of planar maps and graphs
with a condition on the minimum degree, and properties of random graphs from
these classes. In particular we show that the size of the largest tree attached
to the core of a random planar graph is of order c log(n) for an explicit
constant c. These results provide new information on the structure of random
planar graphs.Comment: 32 page
A Potts/Ising Correspondence on Thin Graphs
We note that it is possible to construct a bond vertex model that displays
q-state Potts criticality on an ensemble of phi3 random graphs of arbitrary
topology, which we denote as ``thin'' random graphs in contrast to the fat
graphs of the planar diagram expansion.
Since the four vertex model in question also serves to describe the critical
behaviour of the Ising model in field, the formulation reveals an isomorphism
between the Potts and Ising models on thin random graphs. On planar graphs a
similar correspondence is present only for q=1, the value associated with
percolation.Comment: 6 pages, 5 figure
Random graphs from a weighted minor-closed class
There has been much recent interest in random graphs sampled uniformly from
the n-vertex graphs in a suitable minor-closed class, such as the class of all
planar graphs. Here we use combinatorial and probabilistic methods to
investigate a more general model. We consider random graphs from a
`well-behaved' class of graphs: examples of such classes include all
minor-closed classes of graphs with 2-connected excluded minors (such as
forests, series-parallel graphs and planar graphs), the class of graphs
embeddable on any given surface, and the class of graphs with at most k
vertex-disjoint cycles. Also, we give weights to edges and components to
specify probabilities, so that our random graphs correspond to the random
cluster model, appropriately conditioned.
We find that earlier results extend naturally in both directions, to general
well-behaved classes of graphs, and to the weighted framework, for example
results concerning the probability of a random graph being connected; and we
also give results on the 2-core which are new even for the uniform (unweighted)
case.Comment: 46 page
The Yang Lee Edge Singularity on Feynman Diagrams
We investigate the Yang-Lee edge singularity on non-planar random graphs,
which we consider as the Feynman Diagrams of various d=0 field theories, in
order to determine the value of the edge exponent.
We consider the hard dimer model on phi3 and phi4 random graphs to test the
universality of the exponent with respect to coordination number, and the Ising
model in an external field to test its temperature independence. The results
here for generic (``thin'') random graphs provide an interesting counterpoint
to the discussion by Staudacher of these models on planar random graphs.Comment: LaTeX, 6 pages + 3 figure
Uniform random sampling of planar graphs in linear time
This article introduces new algorithms for the uniform random generation of
labelled planar graphs. Its principles rely on Boltzmann samplers, as recently
developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the
Boltzmann framework, a suitable use of rejection, a new combinatorial bijection
found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic
description of the generating functions counting planar graphs, which was
recently obtained by Gim\'enez and Noy. This gives rise to an extremely
efficient algorithm for the random generation of planar graphs. There is a
preprocessing step of some fixed small cost. Then, the expected time complexity
of generation is quadratic for exact-size uniform sampling and linear for
approximate-size sampling. This greatly improves on the best previously known
time complexity for exact-size uniform sampling of planar graphs with
vertices, which was a little over .Comment: 55 page
Dissections, orientations, and trees, with applications to optimal mesh encoding and to random sampling
We present a bijection between some quadrangular dissections of an hexagon
and unrooted binary trees, with interesting consequences for enumeration, mesh
compression and graph sampling. Our bijection yields an efficient uniform
random sampler for 3-connected planar graphs, which turns out to be determinant
for the quadratic complexity of the current best known uniform random sampler
for labelled planar graphs [{\bf Fusy, Analysis of Algorithms 2005}]. It also
provides an encoding for the set of -edge 3-connected
planar graphs that matches the entropy bound
bits per edge (bpe). This solves a
theoretical problem recently raised in mesh compression, as these graphs
abstract the combinatorial part of meshes with spherical topology. We also
achieve the {optimal parametric rate} bpe
for graphs of with vertices and faces, matching in
particular the optimal rate for triangulations. Our encoding relies on a linear
time algorithm to compute an orientation associated to the minimal Schnyder
wood of a 3-connected planar map. This algorithm is of independent interest,
and it is for instance a key ingredient in a recent straight line drawing
algorithm for 3-connected planar graphs [\bf Bonichon et al., Graph Drawing
2005]
Loop-erased random walk and Poisson kernel on planar graphs
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased
random walk on is . We consider scaling limits
of the loop-erasure of random walks on other planar graphs (graphs embedded
into so that edges do not cross one another). We show that if the
scaling limit of the random walk is planar Brownian motion, then the scaling
limit of its loop-erasure is . Our main contribution is showing
that for such graphs, the discrete Poisson kernel can be approximated by the
continuous one. One example is the infinite component of super-critical
percolation on . Berger and Biskup showed that the scaling limit
of the random walk on this graph is planar Brownian motion. Our results imply
that the scaling limit of the loop-erased random walk on the super-critical
percolation cluster is .Comment: Published in at http://dx.doi.org/10.1214/10-AOP579 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Degree distribution in random planar graphs
We prove that for each , the probability that a root vertex in a
random planar graph has degree tends to a computable constant , so
that the expected number of vertices of degree is asymptotically ,
and moreover that .
The proof uses the tools developed by Gimenez and Noy in their solution to
the problem of the asymptotic enumeration of planar graphs, and is based on a
detailed analysis of the generating functions involved in counting planar
graphs. However, in order to keep track of the degree of the root, new
technical difficulties arise. We obtain explicit, although quite involved
expressions, for the coefficients in the singular expansions of the generating
functions of interest, which allow us to use transfer theorems in order to get
an explicit expression for the probability generating function . From this we can compute the to any degree of accuracy, and derive
the asymptotic estimate for large values of ,
where is a constant defined analytically
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