113 research outputs found
Hamiltonian Cycles on Random Eulerian Triangulations
A random Eulerian triangulation is a random triangulation where an even
number of triangles meet at any given vertex. We argue that the central charge
increases by one if the fully packed O(n) model is defined on a random Eulerian
triangulation instead of an ordinary random triangulation. Considering the case
n -> 0, this implies that the system of random Eulerian triangulations equipped
with Hamiltonian cycles describes a c=-1 matter field coupled to 2D quantum
gravity as opposed to the system of usual random triangulations equipped with
Hamiltonian cycles which has c=-2. Hence, in this case one should see a change
in the entropy exponent from the value gamma=-1 to the irrational value
gamma=(-1-\sqrt{13})/6=-0.76759... when going from a usual random triangulation
to an Eulerian one. A direct enumeration of configurations confirms this change
in gamma.Comment: 22 pages, 9 figures, references and a comment adde
Thoughts on Barnette's Conjecture
We prove a new sufficient condition for a cubic 3-connected planar graph to
be Hamiltonian. This condition is most easily described as a property of the
dual graph. Let be a planar triangulation. Then the dual is a cubic
3-connected planar graph, and is bipartite if and only if is
Eulerian. We prove that if the vertices of are (improperly) coloured blue
and red, such that the blue vertices cover the faces of , there is no blue
cycle, and every red cycle contains a vertex of degree at most 4, then is
Hamiltonian.
This result implies the following special case of Barnette's Conjecture: if
is an Eulerian planar triangulation, whose vertices are properly coloured
blue, red and green, such that every red-green cycle contains a vertex of
degree 4, then is Hamiltonian. Our final result highlights the
limitations of using a proper colouring of as a starting point for proving
Barnette's Conjecture. We also explain related results on Barnette's Conjecture
that were obtained by Kelmans and for which detailed self-contained proofs have
not been published.Comment: 12 pages, 7 figure
Ergodicity of the Wang--Swendsen--Koteck\'y algorithm on several classes of lattices on the torus
We prove the ergodicity of the Wang--Swendsen--Koteck\'y (WSK) algorithm for
the zero-temperature -state Potts antiferromagnet on several classes of
lattices on the torus. In particular, the WSK algorithm is ergodic for
on any quadrangulation of the torus of girth . It is also ergodic for (resp. ) on any Eulerian triangulation of the torus such that
one sublattice consists of degree-4 vertices while the other two sublattices
induce a quadrangulation of girth (resp.~a bipartite quadrangulation)
of the torus. These classes include many lattices of interest in statistical
mechanics.Comment: 27 pages, pdflatex, and 22 pdf figures. Corrected an error in Remark
4 after Theorem 4.4. Final versio
Worm Monte Carlo study of the honeycomb-lattice loop model
We present a Markov-chain Monte Carlo algorithm of "worm"type that correctly
simulates the O(n) loop model on any (finite and connected) bipartite cubic
graph, for any real n>0, and any edge weight, including the fully-packed limit
of infinite edge weight. Furthermore, we prove rigorously that the algorithm is
ergodic and has the correct stationary distribution. We emphasize that by using
known exact mappings when n=2, this algorithm can be used to simulate a number
of zero-temperature Potts antiferromagnets for which the Wang-Swendsen-Kotecky
cluster algorithm is non-ergodic, including the 3-state model on the
kagome-lattice and the 4-state model on the triangular-lattice. We then use
this worm algorithm to perform a systematic study of the honeycomb-lattice loop
model as a function of n<2, on the critical line and in the densely-packed and
fully-packed phases. By comparing our numerical results with Coulomb gas
theory, we identify the exact scaling exponents governing some fundamental
geometric and dynamic observables. In particular, we show that for all n<2, the
scaling of a certain return time in the worm dynamics is governed by the
magnetic dimension of the loop model, thus providing a concrete dynamical
interpretation of this exponent. The case n>2 is also considered, and we
confirm the existence of a phase transition in the 3-state Potts universality
class that was recently observed via numerical transfer matrix calculations.Comment: 33 pages, 12 figure
Precoloring extension in planar near-Eulerian-triangulations
We consider the 4-precoloring extension problem in planar near-Eulerian- triangulations, i.e., plane graphs where all faces except possibly for the outer one have length three, all vertices not incident with the outer face have even degree, and exactly the vertices incident with the outer face are precolored. We give a necessary topological condition for the precoloring to extend, and give a complete characterization when the outer face has length at most five and when all vertices of the outer face have odd degree and are colored using only three colors
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