1,504 research outputs found
On the non-ergodicity of the Swendsen-Wang-Kotecky algorithm on the kagome lattice
We study the properties of the Wang-Swendsen-Kotecky cluster Monte Carlo
algorithm for simulating the 3-state kagome-lattice Potts antiferromagnet at
zero temperature. We prove that this algorithm is not ergodic for symmetric
subsets of the kagome lattice with fully periodic boundary conditions: given an
initial configuration, not all configurations are accessible via Monte Carlo
steps. The same conclusion holds for single-site dynamics.Comment: Latex2e. 22 pages. Contains 11 figures using pstricks package. Uses
iopart.sty. Final version accepted in journa
Sampling Colourings of the Triangular Lattice
We show that the Glauber dynamics on proper 9-colourings of the triangular
lattice is rapidly mixing, which allows for efficient sampling. Consequently,
there is a fully polynomial randomised approximation scheme (FPRAS) for
counting proper 9-colourings of the triangular lattice. Proper colourings
correspond to configurations in the zero-temperature anti-ferromagnetic Potts
model. We show that the spin system consisting of proper 9-colourings of the
triangular lattice has strong spatial mixing. This implies that there is a
unique infinite-volume Gibbs distribution, which is an important property
studied in statistical physics. Our results build on previous work by Goldberg,
Martin and Paterson, who showed similar results for 10 colours on the
triangular lattice. Their work was preceded by Salas and Sokal's 11-colour
result. Both proofs rely on computational assistance, and so does our 9-colour
proof. We have used a randomised heuristic to guide us towards rigourous
results.Comment: 42 pages. Added appendix that describes implementation. Added
ancillary file
On the Parameterized Intractability of Monadic Second-Order Logic
One of Courcelle's celebrated results states that if C is a class of graphs
of bounded tree-width, then model-checking for monadic second order logic
(MSO_2) is fixed-parameter tractable (fpt) on C by linear time parameterized
algorithms, where the parameter is the tree-width plus the size of the formula.
An immediate question is whether this is best possible or whether the result
can be extended to classes of unbounded tree-width. In this paper we show that
in terms of tree-width, the theorem cannot be extended much further. More
specifically, we show that if C is a class of graphs which is closed under
colourings and satisfies certain constructibility conditions and is such that
the tree-width of C is not bounded by \log^{84} n then MSO_2-model checking is
not fpt unless SAT can be solved in sub-exponential time. If the tree-width of
C is not poly-logarithmically bounded, then MSO_2-model checking is not fpt
unless all problems in the polynomial-time hierarchy can be solved in
sub-exponential time
Hyperbolic four-manifolds, colourings and mutations
We develop a way of seeing a complete orientable hyperbolic -manifold
as an orbifold cover of a Coxeter polytope that has a facet colouring. We also develop a way of finding
totally geodesic sub-manifolds in , and describing
the result of mutations along . As an application of our method,
we construct an example of a complete orientable hyperbolic -manifold
with a single non-toric cusp and a complete orientable hyperbolic
-manifold with a single toric cusp. Both and
have twice the minimal volume among all complete orientable
hyperbolic -manifolds.Comment: 24 pages, 11 figures; to appear in Proceedings of the London
Mathematical Societ
Properly coloured copies and rainbow copies of large graphs with small maximum degree
Let G be a graph on n vertices with maximum degree D. We use the Lov\'asz
local lemma to show the following two results about colourings c of the edges
of the complete graph K_n. If for each vertex v of K_n the colouring c assigns
each colour to at most (n-2)/22.4D^2 edges emanating from v, then there is a
copy of G in K_n which is properly edge-coloured by c. This improves on a
result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4),
409-433, 2003]. On the other hand, if c assigns each colour to at most n/51D^2
edges of K_n, then there is a copy of G in K_n such that each edge of G
receives a different colour from c. This proves a conjecture of Frieze and
Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a
framework developed by Lu and Sz\'ekely [Electron. J. Comb. 14(1), R63, 2007]
for applying the local lemma to random injections. In order to improve the
constants in our results we use a version of the local lemma due to Bissacot,
Fern\'andez, Procacci, and Scoppola [preprint, arXiv:0910.1824].Comment: 9 page
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