122 research outputs found
Tiles and colors
Tiling models are classical statistical models in which different geometric
shapes, the tiles, are packed together such that they cover space completely.
In this paper we discuss a class of two-dimensional tiling models in which the
tiles are rectangles and isosceles triangles. Some of these models have been
solved recently by means of Bethe Ansatz. We discuss the question why only
these models in a larger family are solvable, and we search for the Yang-Baxter
structure behind their integrablity. In this quest we find the Bethe Ansatz
solution of the problem of coloring the edges of the square lattice in four
colors, such that edges of the same color never meet in the same vertex.Comment: 18 pages, 3 figures (in 5 eps files
Triangular Trimers on the Triangular Lattice: an Exact Solution
A model is presented consisting of triangular trimers on the triangular
lattice. In analogy to the dimer problem, these particles cover the lattice
completely without overlap. The model has a honeycomb structure of hexagonal
cells separated by rigid domain walls. The transfer matrix can be diagonalised
by a Bethe Ansatz with two types of particles. This leads two an exact
expression for the entropy on a two-dimensional subset of the parameter space.Comment: 4 pages, REVTeX, 5 EPS figure
A Guide to Stochastic Loewner Evolution and its Applications
This article is meant to serve as a guide to recent developments in the study
of the scaling limit of critical models. These new developments were made
possible through the definition of the Stochastic Loewner Evolution (SLE) by
Oded Schramm. This article opens with a discussion of Loewner's method,
explaining how this method can be used to describe families of random curves.
Then we define SLE and discuss some of its properties. We also explain how the
connection can be made between SLE and the discrete models whose scaling limits
it describes, or is believed to describe. Finally, we have included a
discussion of results that were obtained from SLE computations. Some explicit
proofs are presented as typical examples of such computations. To understand
SLE sufficient knowledge of conformal mapping theory and stochastic calculus is
required. This material is covered in the appendices.Comment: 80 pages, 22 figures, LaTeX; this version has 5 minor corrections to
the text and improved hyperref suppor
Exact conjectured expressions for correlations in the dense O loop model on cylinders
We present conjectured exact expressions for two types of correlations in the
dense O loop model on square lattices with periodic
boundary conditions. These are the probability that a point is surrounded by
loops and the probability that consecutive points on a row are on the
same or on different loops. The dense O loop model is equivalent to the
bond percolation model at the critical point. The former probability can be
interpreted in terms of the bond percolation problem as giving the probability
that a vertex is on a cluster that is surrounded by \floor{m/2} clusters and
\floor{(m+1)/2} dual clusters. The conjectured expression for this
probability involves a binomial determinant that is known to give weighted
enumerations of cyclically symmetric plane partitions and also of certain types
of families of nonintersecting lattice paths. By applying Coulomb gas methods
to the dense O loop model, we obtain new conjectures for the asymptotics
of this binomial determinant.Comment: 17 pages, replaced by version accepted by JSTA
Bethe Ansatz solution of triangular trimers on the triangular lattice
Details are presented of a recently announced exact solution of a model
consisting of triangular trimers covering the triangular lattice. The solution
involves a coordinate Bethe Ansatz with two kinds of particles. It is similar
to that of the square-triangle random tiling model, due to M. Widom and P. A.
Kalugin. The connection of the trimer model with related solvable models is
discussed.Comment: 33 pages, LaTeX2e, 13 EPS figures, PSFra
Scaling Properties of the Ising Model in a Field
The dilute A_3 model is a solvable IRF (interaction-round-a-face) model with
three local states and adjacency conditions encoded by the Dynkin diagram of
the Lie algebra A_3. It can be regarded as a solvable version of a critical
Ising model in a magnetic field. One therefore expects the scaling limit to be
governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal
field theory. We perform a detailed numerical investigation of the solutions of
the Bethe ansatz equation for the off-critical model. Our results agree
perfectly with the predicted values for the lowest masses of the stable
particles and support the assumptions on the nature of the Bethe ansatz
solutions which enter crucially in a recent thermodynamic Bethe ansatz
calculation of the factorized scattering matrix.Comment: 10 pages, uuencoded gz-compressed PostScrip
On the integrability of the square-triangle random tiling model
It is shown that the square-triangle random tiling model is equivalent to an
asymmetric limit of the three colouring model on the honeycomb lattice. The
latter model is known to be the O(n) model at T=0 and corresponds to the
integrable model connected to the affine Lie algebra. Thus it is
shown that the weights of the square-triangle random tiling satisfy the
Yang-Baxter equation, albeit in a singular limit of a more general model. The
three colouring model for general vertex weights is solved by algebraic Bethe
Ansatz.Comment: 11 pages, LaTeX, inluding 2 postscript figure
Exact Stationary State for an ASEP with Fully Parallel Dynamics
The exact stationary state of an asymmetric exclusion process with fully
parallel dynamics is obtained using the matrix product Ansatz. We give a simple
derivation for the deterministic case by a physical interpretation of the
dimension of the matrices. We prove the stationarity via a cancellation
mechanism and by making use of an explicit representation of the matrix algebra
we easily find closed expressions for the correlation functions in the general
probabalistic case. Asymptotic expressions, obtained by making use of earlier
results, allow us to derive the exact phase diagram.Comment: 16 pages, 1 figure, corrections to the tex
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